Q.Solve the differential equation $(x\,dy - y\,dx)\,y\sin\left(\frac{y}{x}\right) = (y\,dx + x\,dy)\,x\cos\left(\frac{y}{x}\right)$.

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1. AP EAPCET 2024Set eng-2024-05-20-AN1 markMCQ
   Q.The general solution of the differential equation $\left(x\sin\dfrac{y}{x}\right) dy = \left(y\sin\dfrac{y}{x} - x\right) dx$ is (A) $\cos\dfrac{x}{y} = \log_e x + c$ (B) $\cos\dfrac{y}{x} = \log_e x + c$ (C) $\cos\dfrac{x}{y} = \log_e y + c$ (D) $\cos\dfrac{y}{x} = \log_e y + c$

   ›Reveal solutionSolution

   This is a homogeneous DE; the substitution $y=vx$ cleanly separates variables, giving $\cos(y/x)=\log_e x+c$ — (B).

   Concept and Intuition

   An equation is homogeneous when every term has the same total "degree" once you notice it depends only on the ratio $y/x$. The standard substitution $y=vx$ turns it into a separable equation in $v$ and $x$.

   Step-by-Step Solution

   1. Given: $\left(x\sin\dfrac{y}{x}\right)dy = \left(y\sin\dfrac{y}{x}-x\right)dx$.
   2. Let $y=vx\Rightarrow dy=v\,dx+x\,dv$, and $\dfrac{y}{x}=v$.
   3. Substitute: $x\sin v\,(v\,dx+x\,dv) = (vx\sin v - x)\,dx$.
   4. Expand LHS: $vx\sin v\,dx+x^2\sin v\,dv$. RHS: $vx\sin v\,dx - x\,dx$.
   5. The $vx\sin v\,dx$ term cancels from both sides, leaving: $x^2\sin v\,dv=-x\,dx$.
   6. Divide by $x^2$ (assuming $x\ne0$): $\sin v\,dv=-\dfrac{dx}{x}$.
   7. Integrate both sides: $-\cos v=-\ln|x|+C_1\Rightarrow\cos v=\ln|x|+C$ (relabeling the constant).
   8. Substitute back $v=y/x$: $\cos\dfrac{y}{x}=\log_e x+c$. …

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2. AP EAPCET 2025Set eng-2025-05-27-FN1 markMCQ
   Q.The general solution of the differential equation $\left(x\sin\dfrac{y}{x}\right) dy = \left(y\sin\dfrac{y}{x} - x\right) dx$ is (A) $\cos\left(\dfrac{y}{x}\right) = \log|x| + c$ (B) $\cos\left(\dfrac{y}{x}\right) = \dfrac{1}{x} + c$ (C) $\cos\left(\dfrac{x}{y}\right) = \log|y| + c$ (D) $\cos\dfrac{y}{x} = \dfrac{2}{x} + c$

   ›Reveal solutionSolution

   The equation is homogeneous in $x,y$; the substitution $y=vx$ makes it separable and integrates to $\cos(y/x)=\log|x|+c$.

   Concept and Intuition

   A differential equation where every term has the same total degree in $x,y$ (here, degree 1, since $x\sin(y/x)$ and $y\sin(y/x)$ and $x$ are all "degree 1" once $y/x$ is treated as dimensionless) is homogeneous, and the standard move is $y=vx$: it reduces the two variables $x,y$ to one variable $v=y/x$ plus $x$, making the equation separable.

   Step-by-Step Solution

   1. Given: $(x\sin(y/x))\,dy=(y\sin(y/x)-x)\,dx$.
   2. Let $y=vx$, so $dy=v\,dx+x\,dv$ and $y/x=v$. Substitute:

   $x\sin v\,(v\,dx+x\,dv) = (vx\sin v-x)\,dx.$

   3. Expand: $xv\sin v\,dx+x^2\sin v\,dv = xv\sin v\,dx-x\,dx$.
   4. The $xv\sin v\,dx$ terms on both sides cancel exactly, leaving $x^2\sin v\,dv=-x\,dx \ \Rightarrow\ \sin v\,dv=-\frac{dx}{x}\quad(x\neq0).$ …

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3. AP EAPCET 2025Set eng-2025-05-21-FN1 markMCQ
   Q.The general solution of the differential equation $\left(x\sin\dfrac{y}{x}\right)dy=\left(y\sin\dfrac{y}{x}-x\right)dx$ is (A) $\log x+\tan\dfrac{y}{x}=c$ (B) $\log x+\cos\dfrac{y}{x}=c$ (C) $\log x-\sin\dfrac{y}{x}=c$ (D) $\log x-\cos\dfrac{y}{x}=c$

   ›Reveal solutionSolution

   This is a homogeneous first-order ODE solved via $y=vx$; the answer is (D).

   Concept and Intuition

   When every term in the ODE is a function of $y/x$ times some power of $x$ or $y$ (a homogeneous equation), the substitution $y=vx$ (so $v=y/x$) reduces it to a separable equation in $v$ and $x$.

   Step-by-Step Solution

   1. The given equation is $\left(x\sin\frac yx\right)dy=\left(y\sin\frac yx-x\right)dx$, so

   $\frac{dy}{dx}=\frac{y\sin(y/x)-x}{x\sin(y/x)}.$

   2. Substitute $y=vx$, $\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$. The right side becomes $\dfrac{vx\sin v - x}{x\sin v}=\dfrac{v\sin v-1}{\sin v}=v-\dfrac1{\sin v}$.
   3. So $v+x\dfrac{dv}{dx}=v-\dfrac1{\sin v}$, giving $x\dfrac{dv}{dx}=-\dfrac1{\sin v}$.
   4. Separate variables: $\sin v\,dv=-\dfrac{dx}{x}$. …

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4. AP EAPCET 2024Set eng-2024-05-21-AN1 markMCQ
   Q.$\dfrac{dy}{dx} = \dfrac{y + x\tan\dfrac{y}{x}}{x} \Rightarrow \sin\dfrac{y}{x} =$ (A) $cx^2$ (B) $cx$ (C) $cx^3$ (D) $cx^4$

   ›Reveal solutionSolution

   Substituting $y=vx$ reduces the equation to $\cot v\,dv = dx/x$, giving $\sin(y/x)=cx$.

   Concept and Intuition

   An ODE of the form $dy/dx = F(y/x)$ is homogeneous and is solved by substituting $v=y/x$, which always cancels the $v$ term linearly and leaves a separable equation in $v$ and $x$.

   Step-by-Step Solution

   1. Let $y=vx \Rightarrow \dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$.
   2. RHS: $\dfrac{y+x\tan(y/x)}{x} = v+\tan v$.
   3. So $v+x\dfrac{dv}{dx} = v+\tan v \Rightarrow x\dfrac{dv}{dx}=\tan v$.
   4. Separate: $\cot v\,dv = \dfrac{dx}{x}$.
   5. Integrate: $\log|\sin v| = \log|x|+\text{const} \Rightarrow \sin v = cx$. …

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5. AP EAPCET 2022Set eng-2022-07-06-AN1 markMCQ
   Q.If $\cos\dfrac{y}{x} = A\log x + C$ is the general solution of $\left(x\sin\dfrac{y}{x}\right) dy = \left(y\sin\dfrac{y}{x} - x\right) dx$, then $A =$ (A) 2 (B) 1 (C) -1 (D) -2

   ›Reveal solutionSolution

   The equation is homogeneous; the substitution $y=vx$ reduces it to a separable form whose integration directly gives $\cos(y/x)=\log x + C$, so $A=1$.

   Concept and Intuition

   The differential equation $(x\sin(y/x))dy=(y\sin(y/x)-x)dx$ is homogeneous (every term scales the same way under $x\to\lambda x, y\to\lambda y$), so the standard substitution $y=vx$ (with $v=y/x$) converts it into a separable equation in $v$ and $x$.

   Step-by-Step Solution

   1. Let $y=vx$, so $dy=v\,dx+x\,dv$.
   2. Substitute into $(x\sin v)dy = (vx\sin v - x)dx$: $(x\sin v)(v\,dx+x\,dv) = (vx\sin v-x)dx$.
   3. Expand LHS: $vx\sin v\,dx + x^2\sin v\,dv = vx\sin v\,dx - x\,dx$.
   4. The $vx\sin v\,dx$ terms cancel from both sides: $x^2\sin v\,dv = -x\,dx$.
   5. Divide by $x^2$ (assuming $x\ne0$): $\sin v\,dv = -\dfrac{dx}{x}$.
   6. Integrate both sides: $-\cos v = -\log|x| + C' \Rightarrow \cos v = \log|x| + C$. …

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6. AP EAPCET 2024Set eng-2024-05-20-FN1 markMCQ
   Q.The general solution of the differential equation $(x+y)y\,dx+(y-x)x\,dy=0$ is (A) $x+y\log(cy)=0$ (B) $\dfrac{y}{x}=\log(xy)+c$ (C) $x+y\log(cxy)=0$ (D) $\dfrac{y}{x}=\log(cxy)$

   ›Reveal solutionSolution

   This is a homogeneous first-order ODE; the substitution $y=vx$ turns it into a separable equation whose integral rearranges to $x+y\log(cxy)=0$.

   Concept and Intuition

   Every term of $(x+y)y\,dx+(y-x)x\,dy=0$ is of the same total degree (degree 2 in $x,y$), which is the signature of a homogeneous differential equation. The standard technique is to set $y=vx$, turning the equation into one involving only $v$ and $x$, which separates.

   Step-by-Step Solution

   1. Expand: $(xy+y^2)\,dx+(xy-x^2)\,dy=0$ — every term has degree 2, confirming homogeneity.
   2. Put $y=vx$, so $dy=v\,dx+x\,dv$. Then $xy+y^2=x^2v(1+v)$ and $xy-x^2=x^2(v-1)$.
   3. Substituting: $x^2v(1+v)\,dx+x^2(v-1)(v\,dx+x\,dv)=0$. Dividing by $x^2$: $v(1+v)\,dx+(v-1)v\,dx+x(v-1)\,dv=0$.
   4. Collect the $dx$ coefficient: $v(1+v)+v(v-1)=v[(1+v)+(v-1)]=2v^2$. So $2v^2\,dx+x(v-1)\,dv=0$.
   5. Separate variables: $\dfrac{dx}{x}=\dfrac{(1-v)}{2v^2}\,dv=\left(\dfrac{1}{2v^2}-\dfrac{1}{2v}\right)dv$.
   6. Integrate: $\log x = -\dfrac{1}{2v}-\dfrac12\log v+C$. Multiply by 2: $2\log x+\log v=-\dfrac1v+2C\Rightarrow \log(x^2v)=-\dfrac1v+K$.
   7. Since $x^2v=x^2\cdot\dfrac{y}{x}=xy$ and $\dfrac1v=\dfrac{x}{y}$: $\log(xy)+\dfrac{x}{y}=K$. …

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7. AP EAPCET 2026Set eng-2026-05-18-FN1 markMCQ
   Q.The general solution of the differential equation $y(2x + y)\, dx = x(x + y)\, dy$ is (A) $\log\left(\dfrac{cx^2}{y}\right) + \dfrac{y}{x} = 0$ (B) $\log\left(\dfrac{y}{cx^2}\right) + \dfrac{y}{x} = 0$ (C) $\log(cx^2y^2) + \dfrac{y}{x} = 0$ (D) $\log(cx^2y) + \dfrac{y}{x} = 0$

   ›Reveal solutionSolution

   A homogeneous ODE solved by the substitution $y=vx$; simplifying and integrating gives $\log\left(\dfrac{y}{cx^2}\right)+\dfrac{y}{x}=0$.

   Concept and Intuition

   $y(2x+y)dx=x(x+y)dy$ is homogeneous of degree 2 on both sides, so the standard substitution $y=vx$ (with $dy=v\,dx+x\,dv$) reduces it to a separable equation in $v$ and $x$.

   Step-by-Step Solution

   1. Put $y=vx$, $dy=v\,dx+x\,dv$.
   2. LHS: $y(2x+y)=vx(2x+vx)=vx^2(2+v)$.
   3. RHS: $x(x+y)\,dy=x\cdot x(1+v)(v\,dx+x\,dv)=x^2(1+v)(v\,dx+x\,dv)$.
   4. Equation becomes $vx^2(2+v)\,dx=x^2(1+v)(v\,dx+x\,dv)$. Divide by $x^2$: $v(2+v)dx=v(1+v)dx+x(1+v)dv$.
   5. Subtract $v(1+v)dx$ from both sides: $v[(2+v)-(1+v)]dx=x(1+v)dv \Rightarrow v\,dx=x(1+v)\,dv$.
   6. Separate: $\dfrac{dx}{x}=\dfrac{1+v}{v}dv=\left(\dfrac1v+1\right)dv$.
   7. Integrate: $\log x=\log v+v+C \Rightarrow \log\dfrac{x}{v}-v=C$.
   8. Substitute back $v=y/x$, so $x/v=x^2/y$: $\log\dfrac{x^2}{y}-\dfrac{y}{x}=C$. …

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8. AP EAPCET 2024Set eng-2024-05-22-FN1 markMCQ
   Q.The general solution of the differential equation $x\,dy - y\,dx = \sqrt{x^2 + y^2}\,dx$ is (A) $y + \sqrt{x^2 + y^2} = cx^2$ (B) $y + \sqrt{x^2 + y^2} = cx$ (C) $x + \sqrt{x^2 + y^2} = cy$ (D) $x - \sqrt{x^2 + y^2} = cy^2$

   ›Reveal solutionSolution

   A homogeneous differential equation, solved with the substitution $y=vx$. Answer: $y+\sqrt{x^2+y^2}=cx^2$.

   Concept and Intuition

   Every term here — $x\,dy$, $y\,dx$, and $\sqrt{x^2+y^2}\,dx$ — scales the same way if $x,y$ are both scaled by the same factor, which is the signature of a homogeneous equation. The standard substitution $y=vx$ converts it into a separable equation in $v$ and $x$.

   Step-by-Step Solution

   1. Write $\dfrac{dy}{dx} = \dfrac{y+\sqrt{x^2+y^2}}{x}$ (dividing through by $dx$, taking $x>0$).
   2. Let $y=vx$, so $\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$. Then: $v+x\frac{dv}{dx} = \frac{vx+\sqrt{x^2+v^2x^2}}{x} = v+\sqrt{1+v^2}.$
   3. So $x\dfrac{dv}{dx}=\sqrt{1+v^2}$, which separates: $\dfrac{dv}{\sqrt{1+v^2}}=\dfrac{dx}{x}$.
   4. Integrate: $\log\left(v+\sqrt{1+v^2}\right) = \log x + \log C \Rightarrow v+\sqrt{1+v^2}=Cx$. …

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9. AP EAPCET 2022Set eng-2022-07-08-AN1 markMCQ
   Q.The general solution of xdy - ydx = ydy is (A) $y = Ae^{-x/y}$ (B) $y = Ae^{x}$ (C) $\dfrac{y}{x} = Ae^{x}$ (D) $\dfrac{x}{y} + \dfrac{y}{x} = C$

   ›Reveal solutionSolution

   The ODE is homogeneous; solving with $y=Vx$ and simplifying the resulting logarithmic relation gives $y=Ae^{-x/y}$.

   Concept and Intuition

   "$x\,dy-y\,dx$" style equations are homogeneous in $x,y$ and standardly solved by substituting $y=Vx$, which turns them into separable equations in $V$ and $x$.

   Step-by-Step Solution

   1. Rearrange: $x\,dy-y\,dy=y\,dx \Rightarrow dy(x-y)=y\,dx \Rightarrow \dfrac{dy}{dx}=\dfrac{y}{x-y}$.
   2. Let $y=Vx$, so $\dfrac{dy}{dx}=V+x\dfrac{dV}{dx}$. RHS becomes $\dfrac{Vx}{x-Vx}=\dfrac{V}{1-V}$.
   3. So $V+x\dfrac{dV}{dx}=\dfrac{V}{1-V} \Rightarrow x\dfrac{dV}{dx} = \dfrac{V-V(1-V)}{1-V}=\dfrac{V^2}{1-V}$.
   4. Separate: $\dfrac{1-V}{V^2}dV = \dfrac{dx}{x} \Rightarrow \left(\dfrac1{V^2}-\dfrac1V\right)dV=\dfrac{dx}{x}$.
   5. Integrate: $-\dfrac1V-\log V = \log x+C_1$.
   6. Substitute back $V=y/x$: $-\dfrac{x}{y}-\log\dfrac{y}{x}=\log x+C_1 \Rightarrow -\dfrac{x}{y}-\log y+\log x=\log x+C_1 \Rightarrow -\dfrac{x}{y}-\log y=C_1$. …

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10. AP EAPCET 2026Set eng-2026-05-14-FN1 markMCQ
    Q.The general solution of the differential equation $\dfrac{dy}{dx} = \dfrac{2xy - 3y^2}{2x^2 + 3xy}$ is (A) $3\log\left|\dfrac{y}{x}\right| = \dfrac{x}{y} + c$ (B) $\log|xy| = 2xy + c$ (C) $3\log|xy| = \dfrac{2x}{y} + c$ (D) $\log\left|\dfrac{y}{x}\right| = xy + c$

    ›Reveal solutionSolution

    The DE is homogeneous of degree 0; substituting $y=vx$ and separating variables leads, after converting back to $x,y$, to $3\log|xy|=\dfrac{2x}{y}+c$.

    Concept and Intuition

    A first-order DE $\dfrac{dy}{dx}=\dfrac{P(x,y)}{Q(x,y)}$ is homogeneous when $P$ and $Q$ are both homogeneous of the same degree — here both numerator ($2xy-3y^2$) and denominator ($2x^2+3xy$) are degree 2. The standard technique is the substitution $y=vx$, which converts the equation into one in $v$ and $x$ alone that separates.

    Step-by-Step Solution

    1. Given: $\dfrac{dy}{dx} = \dfrac{2xy-3y^2}{2x^2+3xy}$. Both numerator and denominator are homogeneous of degree 2, so substitute $y=vx$, $\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$.
    2. Divide numerator and denominator by $x^2$:

    $\frac{2xy-3y^2}{2x^2+3xy} = \frac{2v-3v^2}{2+3v}$

    3. So $v + x\dfrac{dv}{dx} = \dfrac{2v-3v^2}{2+3v}$.
    4. Isolate the derivative term:

    $x\frac{dv}{dx} = \frac{2v-3v^2}{2+3v} - v = \frac{2v-3v^2-v(2+3v)}{2+3v}=\frac{2v-3v^2-2v-3v^2}{2+3v} = \frac{-6v^2}{2+3v}$

    5. Separate variables:

    $\frac{2+3v}{v^2}\,dv = \frac{-6}{x}\,dx \;\Longrightarrow\; \left(\frac{2}{v^2}+\frac{3}{v}\right)dv = -\frac{6}{x}\,dx$

    6. Integrate both sides:

    $-\frac{2}{v}+3\log|v| = -6\log|x| + C$

    7. Rearranging: $-\dfrac2v + 3\log|v| + 6\log|x| = C$. Since $6\log|x|=3\log(x^2)$, combine logs: …

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11. AP EAPCET 2025Set eng-2025-05-23-AN1 markMCQ
    Q.The general solution of the differential equation $(x - (x+y)\log(x+y))dx + x\,dy = 0$ is (A) $y\log(x+y) = cx$ (B) $x\log(x+y) = cy$ (C) $\log(x+y) = cy$ (D) $\log(x+y) = cx$

    ›Reveal solutionSolution

    This tests recognizing that regrouping terms via $v=x+y$ converts an awkward mixed equation into a separable one. The general solution is $\log(x+y)=cx$, option (D).

    Concept and Intuition

    The given equation mixes $x$, $y$, and $\log(x+y)$ in a way that isn't obviously separable or linear at first glance. The key insight is spotting that $dx+dy$ (which appears once you regroup) is exactly $d(x+y)$ — so substituting $v=x+y$ collapses the two-variable equation into one purely in $v$ and $x$, which then separates cleanly.

    Step-by-Step Solution

    1. Expand the given equation: $x\,dx-(x+y)\log(x+y)\,dx+x\,dy=0$.
    2. Regroup: $x(dx+dy)=(x+y)\log(x+y)\,dx$.
    3. Let $v=x+y$, so $dv=dx+dy$. The equation becomes $x\,dv=v\log v\,dx$.
    4. Separate variables: $\dfrac{dv}{v\log v}=\dfrac{dx}{x}$.
    5. For the left side, let $w=\log v$, $dw=\dfrac{dv}{v}$, so $\dfrac{dv}{v\log v}=\dfrac{dw}{w}$, integrating to $\log|w|=\log|\log v|$.
    6. Integrate both sides: $\log|\log v|=\log|x|+C_1 \Rightarrow \log v = kx$ for some constant $k$ (absorbing $e^{C_1}$ and sign). …

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12. AP EAPCET 2026Set eng-2026-05-13-AN1 markMCQ
    Q.The general solution of the differential equation $x\left(\dfrac{dy}{dx}\right)^2 + 2\sqrt{xy}\dfrac{dy}{dx} + y = 0$ is (A) $x+y=c$ (B) $\sqrt{x}+\sqrt{y}=\sqrt{c}$ (C) $x^2+y^2=c^2$ (D) $\sqrt{x}-\sqrt{y}=\sqrt{c}$

    ›Reveal solutionSolution

    The equation is a perfect-square quadratic in $dy/dx$ (discriminant zero), which reduces it to a simple separable equation whose solution is $\sqrt x+\sqrt y=\sqrt c$.

    Concept and Intuition

    When a differential equation is quadratic in $p=dy/dx$, check the discriminant. If it is a perfect square (or zero), the quadratic factors nicely, collapsing the equation to a single, much simpler first-order relation that can be solved by separation of variables.

    Step-by-Step Solution

    1. Given: $x\left(\dfrac{dy}{dx}\right)^2 + 2\sqrt{xy}\dfrac{dy}{dx} + y = 0$. Let $p=dy/dx$: $xp^2+2\sqrt{xy}\,p+y=0$.
    2. Treat as a quadratic in $p$: discriminant $= (2\sqrt{xy})^2 - 4(x)(y) = 4xy - 4xy = 0$.
    3. Since the discriminant is zero, $p = \dfrac{-2\sqrt{xy}}{2x} = -\dfrac{\sqrt{xy}}{x} = -\sqrt{\dfrac{y}{x}}$ (using $\sqrt{xy}/x = \sqrt{y}/\sqrt{x}$).
    4. So $\dfrac{dy}{dx} = -\sqrt{\dfrac{y}{x}}$, which separates as $\dfrac{dy}{\sqrt y} = -\dfrac{dx}{\sqrt x}$.
    5. Integrate both sides: $2\sqrt y = -2\sqrt x + C_1 \Rightarrow \sqrt x + \sqrt y = \dfrac{C_1}{2}$. …

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