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Mathematics · Class 12 Science

Ch 12Differential Equations — Class 12 Mathematics, concept-first.

An equation that involves an independent variable, a dependent variable, and one or more derivatives of the dependent variable with respect to the independent variable is called a differential equation.

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Key concepts

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General and Particular Solutions of a Differential Equation

A differential equation typically has infinitely many solutions, all captured at once by its general solution — an expression containing exactly as many independent arbitrary constants as the order of the equation (a fir…

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Chapter contents

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1

Definition of a Differential Equation

An equation that involves an independent variable, a dependent variable, and one or more derivatives of the dependent variable with respect to the independent variable is called a differential equatio…

2

Order and Degree

Two numbers are attached to every differential equation before it is solved: its order and its degree.

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General and Particular Solutions of a Differential Equation

A function is called a solution of a differential equation if, when and its derivatives (computed from ) are substituted into the equation, the equation is satisfied identically -- i.e.

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Solution by Separation of Variables

The simplest first-order, first-degree differential equations to solve are those whose right-hand side, when written as , factorises as a function of alone times a function of alone: Such an equation…

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Homogeneous Differential Equations of First Order and First Degree

Homogeneous function. A function is called a homogeneous function of degree if, for every , For example, satisfies , so it is homogeneous of degree .

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Linear Differential Equations of the Type dy/dx + Py = Q

A first-order differential equation is called linear in if it can be written in the standard form where and are given functions of only (or constants) -- crucially, and each appear to the first power…

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Linear Differential Equations of the Type dx/dy + Px = Q

Not every linear-looking equation is linear in . If, when solved for a derivative, the equation instead has , , or another non-linear combination of and , it may still be linear in , if it can be arra…

Summary

Differential equation. An equation relating an independent variable, a dependent variable, and derivatives of the dependent variable; this chapter treats only ordinary differential equations (one inde…

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 20 questions20 questions
  1. Q1Find the differential equation of the circles which touch the x-axis at the origin.Preview
  2. Q2Solve: (e^x + 1) y dy - (y^2 + 1) e^x dx = 0, given x = 0, y = 0. **OR** Solve: (x dy - y dx) . y sin(y/x) = (y dx + x dy) . x cos(y/x).Preview
  3. Q3Solve (1 + x^2) dy + 2xy dx = cot x dx.Preview
  4. Q4Solve: dy/dx = e^(x-y) + x² e^(-y).Preview
  5. Q5Solve x cos(y/x) dy/dx = y cos(y/x) + x. **OR** Find the equation of the curve passing through the point (-2, 3) given the slope of the tang…Preview
  6. Q6Find differential equation by eliminating a and b from y = e^x (a cos x + b sin x).Preview
  7. Q7The degree of the differential equation d³y/dx³ + y = (1 + dy/dx)^(1/3) is: **OR** The value of ∫ e^(a log_e x) dx will be (a ≠ -1), (c is a…Preview
  8. Q8Order of the differential equation (d²y/dx²)² = 1 + (dy/dx)³ is (a) 1 (b) 3 (c) 2 (d) 4Preview
  9. Q9Find the differential equation of y = eˣ(a+bx) (a, b constant).Preview
  10. Q10Solve the differential equation x²dy + (xy+y²)dx = 0, given x = 1, y = 0. **OR** Solve the differential equation (1+x²)dy/dx + 2xy = 1/(1+x²…Preview
  11. Q11Find the general solution of eˣdy + (eˣ.y + 2x)dx = 0.Preview
  12. Q12Order and degree of the differential equation y = dy/dx + c/(dy/dx) are (a) 1, 2 (b) 2, 2 (c) 1, 1 (d) 2, 1Preview
  13. Q13Solve (x + 2y³) dy/dx = y, given x = 1, when y = -1. **OR** Solve the differential equation x dx + y dy + (x dy - y dx)/(x²+y²) = 0, given y…Preview
  14. Q14Solve (x² + y²) dx - 2xy dy = 0, given y = 0, when x = 1.Preview
  15. Q15The general solution of the differential equation dy/dx = e^(x+y) is (a) eˣ+eʸ=c (b) eˣ+e⁻ʸ=c (c) e⁻ˣ+eʸ=c (d) e⁻ˣ+e⁻ʸ=cPreview
  16. Q16Find the differential equation by eliminating the constants a and b from the equation y = eˣ(a+bx²).Preview
  17. Q17Solve: e⁻ʸ sec²y dy = dx + x dy. **OR** Solve: x²(x dx + y dy) + 2y(x dy - y dx) = 0.Preview
  18. Q18Solve: x dy/dx - y = x tan(y/x); given y=π/2 when x=1.Preview
  19. Q19Find the differential equation of the circles passing through the points of intersection of unit circle with centre at the origin and line b…Preview
  20. Q20Solve: $x\, dy + y\, dx + y^2 (x\, dy - y\, dx) = 0$.Preview

More questions

30 Q
+Show 3 questions3 questions
  1. Q28Form the differential equation representing the family of curves $y = Ae^{2x} + Be^{-3x}$, where $A, B$ are arbitrary constants, and state i…Free
  2. Q29Solve the initial value problem $\dfrac{dy}{dx} + \dfrac{3}{x}y = x$, $x>0$, given that $y(1) = 1$.Preview
  3. Q30Solve the differential equation $(x^2 - y^2)\,dx + 2xy\,dy = 0$.Preview
+Show 8 questions8 questions
  1. Example 1Find the order and degree (if defined) of the differential equation $\left(\dfrac{d^3y}{dx^3}\right)^2 + \left(\dfrac{dy}{dx}\right)^4 = x^2…Free
  2. Example 2Find the order and degree (if defined) of the differential equation $\dfrac{dy}{dx} + \sin\!\left(\dfrac{dy}{dx}\right) = 0$.Free
  3. Example 3Verify that $y = e^{4x}$ is a solution of the differential equation $\dfrac{d^2y}{dx^2} - \dfrac{dy}{dx} - 12y = 0$.Free
  4. Example 4Verify that $y = A\sin x + B\cos x$, where $A, B$ are arbitrary constants, is the general solution of $\dfrac{d^2y}{dx^2} + y = 0$.Preview
  5. Example 5Find the general solution of the differential equation $\dfrac{dy}{dx} = \dfrac{1+y^2}{1+x^2}$.Preview
  6. Example 6Show that the differential equation $(x^2+y^2)\,dx = 2xy\,dy$ is homogeneous, and find its general solution.Preview
  7. Example 7Find the general solution of the differential equation $\dfrac{dy}{dx} + \dfrac{2}{x}y = x^2$, $x>0$.Preview
  8. Example 8Find the general solution of the differential equation $\dfrac{dx}{dy} + \dfrac{x}{y} = y$, $y>0$.Preview
+Show 4 questions4 questions
  1. Q9Find the order and degree of the differential equation $\dfrac{d^2y}{dx^2} + 5\dfrac{dy}{dx} - 6y = \log x$.Free
  2. Q10Find the order and degree of the differential equation $\left(\dfrac{d^3y}{dx^3}\right)^{2/3} = \dfrac{dy}{dx} + y$.Free
  3. Q11Find the order and degree of the differential equation $x^2\left(\dfrac{d^2y}{dx^2}\right)^2 + x\left(\dfrac{dy}{dx}\right)^3 = 0$.Preview
  4. Q12Find the order and degree (if defined) of the differential equation $\dfrac{dy}{dx} = e^{\,d^2y/dx^2}$.Preview
+Show 5 questions5 questions
  1. Q13Solve the differential equation $\dfrac{dy}{dx} = \dfrac{x}{y}$, $y \neq 0$.Free
  2. Q14Solve the differential equation $\dfrac{dy}{dx} = e^{x-y}$.Free
  3. Q15Solve the differential equation $(1+x^2)\,dy + (1+y^2)\,dx = 0$.Preview
  4. Q16Solve the differential equation $x\,dy - y\,dx = 0$, given that $y = 2$ when $x = 1$.Preview
  5. Q17Solve the differential equation $\sec^2x \tan y\,dx + \sec^2y \tan x\,dy = 0$.Preview
+Show 5 questions5 questions
  1. Q18Solve the differential equation $\dfrac{dy}{dx} = \dfrac{y}{x} + \dfrac{x}{y}$.Free
  2. Q19Solve the differential equation $\dfrac{dy}{dx} = \dfrac{2xy}{x^2+y^2}$.Free
  3. Q20Solve the differential equation $\dfrac{dy}{dx} = \dfrac{y^2}{xy - x^2}$.Preview
  4. Q21Solve the differential equation $x^2\,dy + (y^2 - xy)\,dx = 0$.Preview
  5. Q22Solve the differential equation $(x-y)\,dy = (x+y)\,dx$.Preview
+Show 5 questions5 questions
  1. Q23Solve the differential equation $\dfrac{dy}{dx} + 3y = e^{2x}$.Free
  2. Q24Solve the differential equation $x\dfrac{dy}{dx} + y = x^2$, $x>0$.Free
  3. Q25Solve the differential equation $\dfrac{dy}{dx} + y\tan x = \cos x$, $-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$.Preview
  4. Q26Solve the differential equation $\dfrac{dx}{dy} + 2x = y$.Preview
  5. Q27Solve the differential equation $y\dfrac{dx}{dy} - x = y^3$, $y>0$.Preview