Questions

1. Define ordinary differential equation. Write applications of differential equations.

2. If centre of a circle lies on x-axis and the circle passes through origin, form differential equation of that circle.

3. Solve following differential equations by separation of variables:

$$\sin^{-1}\!\left(\frac{dy}{dx}\right) = x + y$$ $$\frac{dy}{dx} = (4x + y + 1)^2$$

4. Find order and degree of following equations:

$$\frac{dy}{dx} = e^x$$ $$\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} - \tan^2 y = 0$$ $$\frac{d^3 y}{dx^3} + \frac{d^2 y}{dx^2} - \frac{dy}{dx} + y = x \ln x$$ $$\left(\frac{d^2 y}{dx^2}\right)^3 + 6\frac{dy}{dx} = 0$$

5. Form differential equation whose solutions are given below, where $A$, $B$, and $C$ are arbitrary constants:

$$y = A e^{2x} + B e^x + C$$ $$y = e^x (A \cos x + B \sin x)$$

6. Solve following exact differential equations:

$$\left(1 + e^{x/y}\right)\,dx + e^{x/y}\left(1 - \frac{x}{y}\right)\,dy = 0$$ $$\left(x^2 y - 2xy^2\right)\,dx - \left(x^3 - 3x^2 y\right)\,dy = 0$$

7. Calculate the particular solution of

$$\frac{d^2 y}{dx^2} + 3\frac{dy}{dx} + 2y = 0$$

when $y(0)=0$ and $y'(0)=1$.

8. Find general solution of following non-homogeneous linear differential equations:

$$(D^2 - 3D + 2)y = 2\cosh x$$ $$(D^2 - 4D + 4)y = x^3$$

9. Define linear differential equation. Find solution of

$$(1 + x^2)\frac{dy}{dx} + y = \tan^{-1} x$$

10. Write Bernoulli's equation. Find solution of

$$\frac{dy}{dx} + \frac{2y}{x} = \frac{y^3}{x^3}$$

using Bernoulli's equation method.

11. Solve the following ODEs:

$$(D^2 - 1)y = 2 \text{ where } y(2)=-1 \text{ and } y'(2)=3$$ $$(D^2 - 4D + 4)y = x^2 \text{ where } x=0,\; y=\frac{3}{8} \text{ and } y'=1$$

12. Find general solution of:

$$\frac{d^2 y}{dx^2} - 3\frac{dy}{dx} + 4y = \cos 4x$$ $$\frac{d^2 y}{dx^2} - 4\frac{dy}{dx} + 4y = x^2 + x$$

13. Find particular solution of

$$\frac{d^2 y}{dx^2} + 4\frac{dy}{dx} + 8y = e^{4x}$$

when $y(0)=0$ and $y'(0)=8$.

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