Q.Find the differential equation of all non-vertical lines in a plane.

Showing the 12 most recent of 21 on this concept.

1. AP EAPCET 2024Set eng-2024-05-23-FN1 markMCQ
   Q.The differential equation for which $ax + by =1$ is general solution is (A) $\dfrac{dy}{dx}=x+c$ (B) $y\dfrac{d^2y}{dx^2}+x=1$ (C) $\dfrac{d^2y}{dx^2}=0$ (D) $\dfrac{d^3y}{dx^3}=0$

   ›Reveal solutionSolution

   Two arbitrary constants means two differentiations to eliminate them; doing so on $ax+by=1$ leaves the trivial equation $y''=0$.

   Concept and Intuition

   To find the differential equation whose general solution is a given family, differentiate the family's equation as many times as there are arbitrary constants, then eliminate those constants using the resulting equations. Here $a,b$ are 2 independent constants, so 2 differentiations (and eliminating $a$ and $b$) should produce a second-order DE free of both constants.

   Step-by-Step Solution

   1. Start with $ax+by=1$.
   2. Differentiate w.r.t. $x$: $a+by'=0$ — (i)
   3. Differentiate (i) again w.r.t. $x$: since $a$ is a constant, $\dfrac{d}{dx}(a)=0$, so $by''=0$.
   4. Since $b$ is (generically) nonzero for a genuine two-parameter family, this forces $y''=0$, i.e. $\dfrac{d^2y}{dx^2}=0$.
   5. This is exactly the differential equation whose general solution is a straight line $y=mx+c$ (equivalent to $ax+by=1$ after rearranging) — consistent with $ax+by=1$ representing a family of straight lines.

   Common Mistakes

   • Stopping after the first differentiation (equation (i)) — that still contains the constant $a$, so it isn't the required DE.
   • Trying to solve for $a$ and $b$ explicitly instead of just eliminating them via successive differentiation.

   ✓Final answer

   The correct option is (C) — $\dfrac{d^2y}{dx^2}=0$.

   ANSWER: C

2. AP EAPCET 2024Set eng-2024-05-22-AN1 markMCQ
   Q.The differential equation of the family of hyperbolas having their centres at origin and their axes along the coordinates axes is (A) $xyy_2+xy_1^2-yy_1=0$ (B) $xy_2-xyy_1^2+yy_1=0$ (C) $xyy_2+xy_1^2+yy_1=0$ (D) $xy_2+xy_1^2-yy_1=0$

   ›Reveal solutionSolution

   Differentiating the two-parameter hyperbola family $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ twice and eliminating $a^2,b^2$ gives $xyy_2+xy_1^2-yy_1=0$, option (A).

   Concept and Intuition

   A family of curves with $n$ independent arbitrary constants satisfies an $n$-th order differential equation obtained by differentiating $n$ times and eliminating the constants. Here the hyperbola family centred at the origin with axes along the coordinate axes has two constants $a,b$, so we need two differentiations.

   Step-by-Step Solution

   1. Family: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$.
   2. Differentiate once: $\dfrac{2x}{a^2}-\dfrac{2yy_1}{b^2}=0\Rightarrow\dfrac{x}{a^2}=\dfrac{yy_1}{b^2}\Rightarrow\dfrac{a^2}{b^2}=\dfrac{x}{yy_1}$.
   3. Differentiate again: $\dfrac{1}{a^2}-\dfrac{(y_1^2+yy_2)}{b^2}=0\Rightarrow\dfrac{1}{a^2}=\dfrac{y_1^2+yy_2}{b^2}$.
   4. Divide the two boxed relations (both equal to a ratio of $1/a^2$ terms with $1/b^2$ factored) to eliminate $a^2,b^2$ entirely:

   $x=\frac{a^2\cdot\frac{yy_1}{b^2}\Big/\frac{1}{a^2}}{\cdots}\ \Rightarrow\ x=\frac{yy_1}{y_1^2+yy_2}$

   (dividing the step-2 relation by the step-3 relation directly cancels $a^2/b^2$).

   5. Cross-multiplying: $x(y_1^2+yy_2)=yy_1\Rightarrow xy_1^2+xyy_2-yy_1=0$, i.e. $xyy_2+xy_1^2-yy_1=0$.

   Common Mistakes

   • Sign error when isolating $a^2/b^2$ from the two differentiated equations.
   • Confusing this with the ellipse family (same steps, but the ellipse equation has a $+$ between the terms, and its DE differs only in the sign in front of $yy_1$ in some derivations — always redo the elimination rather than recalling a memorized DE).

   ✓Final answer

   The correct option is (A) — $xyy_2+xy_1^2-yy_1=0$.

   ANSWER: A

3. AP EAPCET 2022Set eng-2022-07-07-FN1 markMCQ
   Q.The differential equation of the family of circles passing through (0, 0) and having centre on x-axis is (A) $2xy\dfrac{dy}{dx} + x^2 - y^2 = 0$ (B) $\left(\dfrac{dy}{dx}\right)^2 + y\dfrac{d^2y}{dx^2} + 1 = 0$ (C) $xy\dfrac{dy}{dx} + y^2 - x^2 = 0$ (D) $\dfrac{dy}{dx} = \dfrac{x+y}{x-y}$

   ›Reveal solutionSolution

   Eliminating the one-parameter family's constant $h$ (the centre's x-coordinate) between the circle equation and its derivative gives $2xyy'+x^2-y^2=0$.

   Concept and Intuition

   A family of circles through the origin with centres on the x-axis is a one-parameter family (parameter $h$, the centre). To get its differential equation, differentiate the family once (matching the one free parameter) and eliminate $h$ between the original equation and the derived one.

   Step-by-Step Solution

   1. Circle centred at $(h,0)$ through origin: radius $=$ distance from centre to origin $=h$. Equation: $(x-h)^2+y^2=h^2\Rightarrow x^2+y^2-2hx=0$.
   2. Differentiate w.r.t. $x$: $2x+2yy'-2h=0\Rightarrow h=x+yy'$.
   3. From the original equation, $h=\dfrac{x^2+y^2}{2x}$.
   4. Equate: $x+yy'=\dfrac{x^2+y^2}{2x}$. Multiply by $2x$: $2x^2+2xyy'=x^2+y^2$.
   5. Rearrange: $2xyy'=y^2-x^2 \Rightarrow 2xyy'+x^2-y^2=0$.

   Common Mistakes

   • Sign error rearranging step 5 (giving $y^2-x^2$ instead of $x^2-y^2$ on the left).
   • Trying to eliminate two constants (radius and centre independently) — here they're linked ($r=h$) since the circle passes through the origin, so only one true parameter $h$ exists.

   ✓Final answer

   The correct option is (A) — $2xy\dfrac{dy}{dx}+x^2-y^2=0$.

   ANSWER: A

4. AP EAPCET 2025Set eng-2025-05-22-AN1 markMCQ
   Q.The differential equation of the family of circles passing through the origin and having centre on X-axis is (A) $(y^2+x^2)dx - 2ydy = 0$ (B) $(y^2-x^2)dx - 2xydy = 0$ (C) $(y^2-x^2)dx + 2ydy = 0$ (D) $(y^2+x^2)dx + 2ydy = 0$

   ›Reveal solutionSolution

   Eliminating the one parameter $a$ (the centre's $x$-coordinate) from the circle's equation via differentiation gives the differential equation $(y^2-x^2)dx-2xy\,dy=0$.

   Concept and Intuition

   A one-parameter family of curves satisfies a first-order differential equation obtained by differentiating the family's equation once and eliminating the parameter. Here the family is "circles through the origin with centre on the x-axis," which has exactly one parameter: the centre's $x$-coordinate $a$ (the radius must equal $a$ since the circle passes through the origin).

   Step-by-Step Solution

   1. Circle with centre $(a,0)$, radius $a$ (so it passes through the origin): $(x-a)^2+y^2=a^2\Rightarrow x^2+y^2=2ax$.
   2. Differentiate w.r.t. $x$: $2x+2yy'=2a\Rightarrow a=x+yy'$.
   3. Substitute back into $x^2+y^2=2ax$: $x^2+y^2=2x(x+yy')=2x^2+2xyy'$.
   4. Rearranging: $y^2-x^2=2xyy'=2xy\dfrac{dy}{dx}$, i.e. $(y^2-x^2)dx-2xy\,dy=0$.

   Common Mistakes

   • Sign error when moving terms across, e.g. writing $(y^2+x^2)$ instead of $(y^2-x^2)$.
   • Forgetting that "passes through the origin" forces radius $=$ distance from centre to origin $=|a|$, not an independent constant.

   ✓Final answer

   The correct option is (B) — $(y^2-x^2)dx-2xydy=0$.

   ANSWER: B

5. AP EAPCET 2025Set eng-2025-05-21-FN1 markMCQ
   Q.The differential equation corresponding to the family of parabolas whose axis is along $x = 1$ is (A) $\dfrac{d^2y}{dx^2}-(x-1)\dfrac{dy}{dx}=0$ (B) $(x-1)\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}=0$ (C) $\dfrac{d^2y}{dx^2}+(x-1)\dfrac{dy}{dx}-y=0$ (D) $(x-1)\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}=0$

   ›Reveal solutionSolution

   This tests forming a differential equation by eliminating arbitrary constants from a family of curves; the answer is (B).

   Concept and Intuition

   A family of parabolas with a fixed vertical axis $x=1$ has the general equation $y=a(x-1)^2+b$, where $a$ (controls width/orientation) and $b$ (vertical shift of vertex) are the two free parameters. Since there are 2 arbitrary constants, we need a 2nd-order ODE to eliminate them completely.

   Step-by-Step Solution

   1. Write the family: $y=a(x-1)^2+b$.
   2. Differentiate once: $\dfrac{dy}{dx}=2a(x-1)$.
   3. Differentiate again: $\dfrac{d^2y}{dx^2}=2a$, so $a=\dfrac12\dfrac{d^2y}{dx^2}$.
   4. Substitute this back into the first derivative relation: $\dfrac{dy}{dx}=\left(\dfrac{d^2y}{dx^2}\right)(x-1)$.
   5. Rearranging: $(x-1)\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}=0$, which eliminates both $a$ and $b$ (note $b$ already dropped out automatically after the first derivative).

   Common Mistakes

   • Assuming the parabola's equation is $(y-k)^2=4a(x-1)$ (axis horizontal) instead of vertex form with vertical axis — the problem states the axis is the line $x=1$, meaning the axis of symmetry is vertical, so the parabola opens up/down.
   • Not differentiating enough times to eliminate all constants.

   ✓Final answer

   The correct option is (B) — $(x-1)\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}=0$.

   ANSWER: B

6. AP EAPCET 2024Set eng-2024-05-18-FN1 markMCQ
   Q.The differential equation representing the family of circles having their centres on Y-axis is $\left(y_1=\dfrac{dy}{dx} \text{ and } y_2=\dfrac{d^2y}{dx^2}\right)$ (A) $y_2=y(y_1^2+1)$ (B) $y_2=xy(y_1^2+1)$ (C) $xy_2=y_1(y_1^2+1)$ (D) $xy_2=y(y_1^2+1)$

   ›Reveal solutionSolution

   Eliminating the two constants $k$ (center) and $a$ (radius) from $x^2+(y-k)^2=a^2$ via two differentiations yields $xy_2=y_1(y_1^2+1)$.

   Concept and Intuition

   A family of curves with $n$ independent arbitrary constants satisfies a differential equation of order $n$ obtained by differentiating $n$ times and eliminating the constants. Circles centered anywhere on the Y-axis have two free parameters — the center's y-coordinate $k$ and the radius $a$ — so we need exactly two differentiations.

   Step-by-Step Solution

   1. General equation of a circle with center $(0,k)$ and radius $a$: $x^2+(y-k)^2=a^2$.
   2. Differentiate once w.r.t. $x$: $2x+2(y-k)y_1=0\Rightarrow x+(y-k)y_1=0\Rightarrow (y-k)=-\dfrac{x}{y_1}$.
   3. Differentiate again: $1+y_1\cdot y_1+(y-k)y_2=0$, i.e. $1+y_1^2+(y-k)y_2=0$.
   4. Substitute $(y-k)=-x/y_1$ from step 2: $1+y_1^2-\dfrac{x}{y_1}y_2=0$.
   5. Multiply through by $y_1$: $y_1+y_1^3-xy_2=0$, i.e. $xy_2=y_1+y_1^3=y_1(y_1^2+1)$.

   Common Mistakes

   • Forgetting the circle's radius $a$ is also arbitrary (only center is fixed on the Y-axis), which is why two differentiations (not one) are needed.
   • Losing the factor of $y_1$ when multiplying through, leading to a mismatched power in the final answer.

   ✓Final answer

   The correct option is (C) — $xy_2=y_1(y_1^2+1)$.

   ANSWER: C

7. AP EAPCET 2026Set eng-2026-05-15-AN1 markMCQ
   Q.The differential equation obtained by eliminating $A$ and $B$ from the equation $y = A(x+B)^2$ which represents a family of curves (A) $2yy'' = (y')^2$ (B) $yy'' = 2y'$ (C) $2yy'' = y' + y$ (D) $2yy'' = y' - y$

   ›Reveal solutionSolution

   Differentiating the two-parameter family $y=A(x+B)^2$ twice and eliminating $A,B$ algebraically yields the differential equation $2yy''=(y')^2$.

   Concept and Intuition

   A family of curves with $n$ arbitrary constants generally satisfies an $n$th-order differential equation obtained by differentiating $n$ times and eliminating the constants. Here there are two constants ($A$ and $B$), so we differentiate twice and use the original equation plus its first and second derivatives to eliminate both.

   Step-by-Step Solution

   1. Given $y=A(x+B)^2$.
   2. Differentiate once: $y'=2A(x+B)$.
   3. Differentiate again: $y''=2A$, so $A=\dfrac{y''}{2}$.
   4. From step 2, solve for $(x+B)$: $(x+B)=\dfrac{y'}{2A}=\dfrac{y'}{y''}$ (substituting $2A=y''$).
   5. Substitute both $A$ and $(x+B)$ back into the original equation: $y=A(x+B)^2=\dfrac{y''}{2}\cdot\left(\dfrac{y'}{y''}\right)^2=\dfrac{(y')^2}{2y''}$.
   6. Rearranged: $2y\,y''=(y')^2$.

   Common Mistakes

   • Trying to eliminate the constants without using all three of $y,y',y''$ together (needing exactly two differentiations for two constants).
   • Sign or algebra slip when substituting $(x+B)=y'/y''$ back into the squared term.

   ✓Final answer

   The correct option is (A) — $2yy'' = (y')^2$.

   ANSWER: A

8. AP EAPCET 2025Set eng-2025-05-23-FN1 markMCQ
   Q.The differential equation for which $y^2=4a(x+a)$ (a is the parameter) is the general solution is (A) $y=2x\dfrac{dy}{dx}+y\left(\dfrac{dy}{dx}\right)^2$ (B) $y=y\dfrac{dy}{dx}-x\left(\dfrac{dy}{dx}\right)^2$ (C) $x=3\dfrac{dy}{dx}+y\left(\dfrac{dy}{dx}\right)^2$ (D) $y=3x^2\dfrac{dy}{dx}+y^2\left(\dfrac{dy}{dx}\right)^2$

   ›Reveal solutionSolution

   This tests eliminating the arbitrary constant from a one-parameter family to get its differential equation. Differentiate once, solve for $a$, substitute back. Answer: (A).

   Concept and Intuition

   A family of curves with $n$ independent parameters satisfies a differential equation of order $n$. Here only $a$ is a parameter, so one differentiation should let us eliminate it completely and land back on a relation purely in $x, y, y'$.

   Step-by-Step Solution

   1. Start with $y^2 = 4a(x+a) = 4ax + 4a^2$.
   2. Differentiate both sides with respect to $x$ (treating $a$ as constant):

   $2y\frac{dy}{dx} = 4a \implies a = \frac{1}{2}y\frac{dy}{dx}.$

   3. Substitute this expression for $a$ back into the original equation to eliminate $a$:

   $y^2 = 4x\left(\frac{1}{2}yy'\right) + 4\left(\frac{1}{2}yy'\right)^2 = 2xyy' + y^2(y')^2.$

   4. Divide throughout by $y$ (valid away from $y=0$):

   $y = 2x\frac{dy}{dx} + y\left(\frac{dy}{dx}\right)^2.$

   5. This matches option (A) exactly.

   Common Mistakes

   • Forgetting to substitute $a$ back into the original equation and instead trying to differentiate again (that would raise the order unnecessarily since there's only one parameter).
   • Arithmetic slip while squaring $\frac{1}{2}yy'$ — it gives $\frac{1}{4}y^2y'^2$, and multiplying by 4 gives $y^2y'^2$, not $2y^2y'^2$.

   ✓Final answer

   The correct option is (A) — $y=2x\dfrac{dy}{dx}+y\left(\dfrac{dy}{dx}\right)^2$.

   ANSWER: A

9. AP EAPCET 2026Set eng-2026-05-14-FN1 markMCQ
   Q.If $a$ and $h$ are arbitrary constants, then the differential equation corresponding to the family of curves $ax^2 + 2hxy = 1$ is (A) $x^2\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx} + y = 0$ (B) $x^2\dfrac{d^2y}{dx^2} - x\dfrac{dy}{dx} + y = 0$ (C) $x^2\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx} - y = 0$ (D) $x^2\dfrac{d^2y}{dx^2} - x\dfrac{dy}{dx} - y = 0$

   ›Reveal solutionSolution

   With two arbitrary constants $a,h$, we differentiate the family twice and eliminate both constants between the resulting equations, landing on $x^2y''+xy'-y=0$.

   Concept and Intuition

   A family of curves with $n$ independent arbitrary constants satisfies a differential equation of order $n$ (generically), obtained by differentiating the family $n$ times and eliminating the constants among the original equation and its derivatives. Here there are two constants ($a$ and $h$), so we expect (and need) a second-order ODE.

   Step-by-Step Solution

   1. Start with $ax^2 + 2hxy = 1$.
   2. Differentiate once w.r.t. $x$:

   $2ax + 2h(xy'+y) = 0 \;\Longrightarrow\; ax + h(xy'+y) = 0 \quad \text{...(I)}$

   3. Differentiate (I) again w.r.t. $x$:

   $a + h\big(xy''+y'+y'\big) = 0 \;\Longrightarrow\; a + h(xy''+2y') = 0 \quad \text{...(II)}$

   4. From (I): $a = -\dfrac{h(xy'+y)}{x}$ (for $x\neq0$).
   5. Substitute into (II):

   $-\frac{h(xy'+y)}{x} + h(xy''+2y') = 0$

   6. Divide through by $h$ (assuming $h\neq0$) and multiply by $x$:

   $-(xy'+y) + x^2y''+2xy' = 0$

   7. Simplify:

   $x^2y'' + 2xy' - xy' - y = 0 \;\Longrightarrow\; x^2y'' + xy' - y = 0$

   Common Mistakes

   • Stopping after only one differentiation, which still has two unknowns ($a,h$) and a leftover $y'$ — a first-order relation cannot represent a two-parameter family exactly.
   • Sign errors when eliminating $a$ between equations (I) and (II) — carefully substitute rather than subtract term-by-term.

   ✓Final answer

   The correct option is (C) — $x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}-y=0$.

   ANSWER: C

10. AP EAPCET 2025Set eng-2025-05-26-AN1 markMCQ
    Q.If a and b are arbitrary constants, then the differential equation corresponding to the family of curves $y = \tan(ax+b)$ is (A) $(1+x^2)y_2 - 2yy_1 + y = 0$ (B) $(1+y^2)y_2 - 2yy_1^2 = 0$ (C) $(1+x^2)y_2 + 2yy_1^2 = 0$ (D) $(1+y^2)y_2 - 2yy_1^2 + y = 0$

    ›Reveal solutionSolution

    Differentiating $y=\tan(ax+b)$ twice and eliminating the arbitrary constant $a$ (using $\sec^2=1+\tan^2$) directly gives the second-order ODE $(1+y^2)y_2-2yy_1^2=0$.

    Concept and Intuition

    A 2-parameter family of curves satisfies a second-order differential equation obtained by differentiating twice and eliminating both arbitrary constants. Since $b$ only shifts the argument (it never appears explicitly after one differentiation of $\tan$), only $a$ needs eliminating here.

    Step-by-Step Solution

    1. $y=\tan(ax+b)$. First derivative: $y_1=a\sec^2(ax+b)=a(1+\tan^2(ax+b))=a(1+y^2)$.
    2. So $a=\dfrac{y_1}{1+y^2}$ — note $b$ has already dropped out.
    3. Differentiate $y_1=a(1+y^2)$ again with respect to $x$: $y_2=a\cdot2yy_1$ (since $a$ is constant).
    4. Substitute $a=\dfrac{y_1}{1+y^2}$: $y_2=\dfrac{y_1}{1+y^2}\cdot2yy_1=\dfrac{2yy_1^2}{1+y^2}$.
    5. Rearranged: $(1+y^2)y_2=2yy_1^2$, i.e. $(1+y^2)y_2-2yy_1^2=0$.

    Common Mistakes

    • Trying to eliminate $b$ separately — it's automatically gone once you differentiate $\tan(ax+b)$ once (the derivative only involves $a$ and $y$).
    • Confusing this with the $(1+x^2)$ (variable $x$) forms in the distractor options — the elimination naturally produces $(1+y^2)$, not $(1+x^2)$.

    ✓Final answer

    The correct option is (B) — $(1+y^2)y_2 - 2yy_1^2 = 0$.

    ANSWER: B

11. AP EAPCET 2023Set eng-2023-05-16-AN1 markMCQ
    Q.The differential equation having $y = (a+b)e^{cx+d}$ as its general solution, where a, b, c, d are arbitrary constants, is (A) $y^{(4)} + 3yy^{(3)} + 6y^{(2)}y^2 + y = 0$ (B) $y^{(3)} + 4yy^{(2)} + 6y^2 y^{(1)} + 12y = 0$ (C) $y^{(1)} - y = 0$ (D) $yy^{(2)} - \left(y^{(1)}\right)^2 = 0$

    ›Reveal solutionSolution

    Although the solution is written with four letters $a,b,c,d$, only two of them are independent; eliminating them shows the family satisfies the second-order equation $yy''=(y')^2$.

    Concept and Intuition

    The number of arbitrary constants that genuinely matter tells you the order of the corresponding differential equation. Here $a$ and $b$ only ever appear as the sum $a+b$, and $d$ only ever appears multiplying through $e^d$ — a constant factor. So the family $y=(a+b)e^{cx+d}$ is really just $y = Be^{cx}$ with two truly independent constants $B$ and $c$, meaning we should expect (and can derive) a second-order differential equation.

    Step-by-Step Solution

    1. Simplify: $y = (a+b)e^{cx+d} = \left[(a+b)e^d\right]e^{cx} = Be^{cx}$, where $B$ is one constant.
    2. Differentiate: $y' = Bc\,e^{cx} = c\left(Be^{cx}\right) = cy$, so $c = \dfrac{y'}{y}$.
    3. Differentiate again: $y'' = c\,y'$ (since $c$ is constant, differentiating $y'=cy$ gives $y''=cy'$).
    4. Substitute $c = y'/y$: $y'' = \dfrac{y'}{y}\cdot y' = \dfrac{(y')^2}{y}$.
    5. Rearranged: $y\,y'' - (y')^2 = 0$, i.e. $yy^{(2)} - \left(y^{(1)}\right)^2=0$.

    Common Mistakes

    • Treating $a,b,c,d$ as four independent constants and expecting a fourth-order equation — but $a+b$ collapses to one constant and $d$ merges into it via $e^d$, so the true order is 2.
    • Sign errors while eliminating $c$.

    ✓Final answer

    The correct option is (D) — $yy^{(2)} - \left(y^{(1)}\right)^2 = 0$.

    ANSWER: D

12. AP EAPCET 2026Set eng-2026-05-12-AN1 markMCQ
    Q.The differential equation of the family of curves given by $y = e^{3x}(Ax + B)$ where A, B are arbitrary constants, is (A) $\dfrac{d^2y}{dx^2} + 6\dfrac{dy}{dx} + 9y = 0$ (B) $\dfrac{d^2y}{dx^2} + 6\dfrac{dy}{dx} - 9y = 0$ (C) $\dfrac{d^2y}{dx^2} - 6\dfrac{dy}{dx} - 9y = 0$ (D) $\dfrac{d^2y}{dx^2} - 6\dfrac{dy}{dx} + 9y = 0$

    ›Reveal solutionSolution

    This tests recognizing that $y=(Ax+B)e^{kx}$ is the solution family for a repeated root $k,k$, so the ODE is $y''-2ky'+k^2y=0$; here $k=3$ gives $y''-6y'+9y=0$.

    Concept and Intuition

    When a linear second-order ODE with constant coefficients has a repeated characteristic root $m=k$, its general solution is $y=(Ax+B)e^{kx}$ — the extra factor of $x$ compensates for the root being repeated. Recognizing this pattern means we don't need to eliminate constants by differentiating twice; we can go straight from the solution form to the characteristic equation.

    Step-by-Step Solution

    1. The given family $y=e^{3x}(Ax+B)$ matches the repeated-root pattern with $k=3$.
    2. The characteristic equation for a repeated root $m=3$ is $(m-3)^2=0$, i.e. $m^2-6m+9=0$.
    3. Translating back to the differential equation: $\dfrac{d^2y}{dx^2}-6\dfrac{dy}{dx}+9y=0$.
    4. Verification by direct differentiation: $y'=e^{3x}(3Ax+3B+A)$, $y''=e^{3x}(9Ax+9B+6A)$. Then $y''-6y'+9y = e^{3x}\big[(9Ax+9B+6A)-6(3Ax+3B+A)+9(Ax+B)\big]=e^{3x}[0]=0$. Confirmed.

    Common Mistakes

    • Mixing up the sign of the middle term (choosing $+6y'$ or $-9y$) — always verify by substitution as in step 4.
    • Assuming $A,B$ are the roots rather than arbitrary constants of a repeated-root solution.

    ✓Final answer

    The correct option is (D) — $\dfrac{d^2y}{dx^2} - 6\dfrac{dy}{dx} + 9y = 0$.

    ANSWER: D