Differential Equation Practice Problems

Page for Differential Equation practice problems. Here, a list of practice problems will be given for the course on Differential Equation (Exercise).

Ordinary Differential Equation (ODE) Problems

Q1: Find the order and degree of

$\left(\dfrac{d^3y}{dx^3} \right)^5+x^2y \left( \dfrac{d^4y}{dx^4} \right)^2=\sin(\ln x)$.

Q2: Find the value of $m$ for which the differential equation

$\left(xy^2+mx^2y\right)dx + \left(x+y\right)x^2dy=0$

is exact and solve it for this value of $m$.

Q3: Eliminate A and B from to form an ODE, where A and B are constants.

1. y = e^-x(A + Bx)
2. y = A cosx + B sinx
3. y = Ae^x + Be^-x

Q4:

1. Evaluate $\dfrac{1}{D-2}e^{2x}$.
2. Show the set {cos2x, sin2x} is linearly independent.

Q5: Solve the following differential equations.

1. $\dfrac{dy}{dx}+\dfrac{y}{x}= \dfrac{y^2}{x^2}$ [Full Solution]
2. $\dfrac{dy}{dx}+y \tan x= y^3 \sec x$ [Full Solution]
3. $\dfrac{dy}{dx}+\dfrac{y}{x}=x^2 y^6$ [Full Solution]

Q5. Find the complementary function and the particular integral of [How to Find Particular Integral]

(i) $\dfrac{d^2y}{dx^2}+4y= \sin^2{x}$. (ii) $\dfrac{d^2y}{dx^2}+2 \dfrac{dy}{dx}+y=x^2+e^{-x}$ (iii) $\dfrac{d^2y}{dx^2}-4 \dfrac{dy}{dx}+4y=xe^{2x}$.

Q6: Using variation of parameters, solve the differential equation

1. $\dfrac{d^2y}{dx^2}+y= \text{cosec}~x$
2. $\dfrac{d^2y}{dx^2}+4y= \sec 2x$

Q7: Solve the Cauchy-Euler differential equation

$x^2 \dfrac{d^2y}{dx^2}-3x \dfrac{dy}{dx}+4y=\cos(\ln x)$

Q9: (a) Show the function f(x,y) = xy² satisfies the Lipschitz condition on the rectangle |x| ≤ 1 and |y| ≤ 1.

(b) Find the interval in which the initial value problem (IVP)

$\dfrac{dy}{dx}=y^2, \quad y(1)=1$

has a unique solution.

Q10: Problem on Wronskian.

Partial Differential Equation (PDE) Problems

Q1: Discuss the types of a first order PDE with examples.

Q2: Solve x(y-z)p + y(z-x)q = z(x-y) using Lagrange’s method.

Q3: Find a complete integral of

1. u = pq
2. pq = xy
3. pq = 1

Q4: Find the values of $x$ and $y$ for which the partial differential equation $y\dfrac{\partial^2u}{\partial x^2} – x\dfrac{\partial^2u}{\partial y^2}$ is elliptic, hyperbolic and parabolic.

Applications of Differential Equation

Q1: Define an orthogonal trajectory for a family of curves. Find the orthogonal trajectory of the family of circles:

x²+y² = c² where c is an arbitrary constant.

Q2: The coordinates of a moving particle are given by $x=t^2$ and $y=t^3$. Find the velocity and acceleration of the particle when $t=1$.