GATE Mathematics 100 practice problems

GATE Problem Bank - Ordinary Differential Equations (ODE)

Problem 1: First Order Linear ODE

Question:

\[ \frac{dy}{dx}+2y=e^{-x} \]

Step 1: Compare with standard form

\[ \frac{dy}{dx}+P(x)y=Q(x) \] Here, \[ P(x)=2 \] Integrating factor: \[ IF=e^{\int2dx}=e^{2x} \]

Step 2: Multiply by IF

\[ e^{2x}\frac{dy}{dx}+2e^{2x}y=e^x \] Left side: \[ \frac{d}{dx}(ye^{2x})=e^x \]

Step 3: Integrate

\[ ye^{2x}=e^x+C \] Therefore:

\[ \boxed{ y=e^{-x}+Ce^{-2x} } \]

Problem 2: Separable Differential Equation

Question:

\[ \frac{dy}{dx}=xy \]

Step 1: Separate variables

\[ \frac{dy}{y}=x dx \]

Step 2: Integrate

\[ \ln y=\frac{x^2}{2}+C \] Taking exponential: \[ y=Ce^{x^2/2} \]

\[ \boxed{ y=Ce^{x^2/2} } \]

Problem 3: Exact Differential Equation

Question:

\[ (2xy+y^2)dx+(x^2+2xy)dy=0 \] Let: \[ M=2xy+y^2 \] \[ N=x^2+2xy \] Check exactness: \[ \frac{\partial M}{\partial y} = 2x+2y \] \[ \frac{\partial N}{\partial x} = 2x+2y \] Therefore exact.

Integrate M with respect to x

\[ \int(2xy+y^2)dx \] \[ =x^2y+xy^2+\phi(y) \] Differentiate with respect to y: \[ x^2+2xy+\phi'(y) \] Compare with N: \[ \phi'(y)=0 \] Therefore:

\[ \boxed{ x^2y+xy^2=C } \]

Problem 4: Second Order Homogeneous ODE

Question:

\[ y''-5y'+6y=0 \]

Step 1: Auxiliary equation

\[ m^2-5m+6=0 \] \[ (m-2)(m-3)=0 \] Roots: \[ m=2,3 \]

\[ \boxed{ y=C_1e^{2x}+C_2e^{3x} } \]

Problem 5: Non-Homogeneous ODE (CF + PI)

Question:

\[ y''-4y'+4y=e^{2x} \]

CF

Auxiliary equation: \[ m^2-4m+4=0 \] \[ (m-2)^2=0 \] Therefore: \[ CF=(C_1+C_2x)e^{2x} \]

PI

Since \(e^{2x}\) repeats in CF, multiply trial by \(x^2\). Assume: \[ PI=Ax^2e^{2x} \] Substitution gives: \[ A=\frac12 \] Therefore:

\[ \boxed{ y=(C_1+C_2x)e^{2x} +\frac12x^2e^{2x} } \]

Problem 6: Cauchy-Euler Equation

Question:

\[ x^2y''-3xy'+4y=0 \] Assume: \[ y=x^m \] Derivatives: \[ xy'=mx^m \] \[ x^2y''=m(m-1)x^m \] Substitute: \[ m(m-1)-3m+4=0 \] \[ m^2-4m+4=0 \] \[ (m-2)^2=0 \]

\[ \boxed{ y=x^2(C_1+C_2\ln x) } \]

Problem 7: Laplace Transform ODE

Question:

\[ y'+3y=0 \] Given: \[ y(0)=5 \] Laplace: \[ sY(s)-5+3Y(s)=0 \] Therefore: \[ Y(s)=\frac5{s+3} \] Inverse:

\[ \boxed{ y=5e^{-3t} } \]

Problem 8: Second Order Initial Value Problem

Question:

\[ y''+4y=0 \] Given: \[ y(0)=1,\quad y'(0)=0 \] Solution form: \[ y=C_1\cos2x+C_2\sin2x \] Apply conditions: \[ C_1=1 \] \[ C_2=0 \]

\[ \boxed{ y=\cos2x } \]

Problem 9: Bernoulli Equation

Question:

\[ y'+y=y^2 \] Divide by \(y^2\): \[ y^{-2}y'+y^{-1}=1 \] Let: \[ v=y^{-1} \] Then: \[ v'-v=-1 \] Solving: \[ v=1+Ce^x \] Since: \[ v=\frac1y \]

\[ \boxed{ y=\frac1{1+Ce^x} } \]

Problem 10: Linear Differential Operator

Question:

\[ (D^2-3D+2)y=e^x \]

CF

\[ m^2-3m+2=0 \] \[ m=1,2 \] \[ CF=C_1e^x+C_2e^{2x} \]

PI

Because RHS \(e^x\) is repeated: \[ PI=Axe^x \] Substitution gives: \[ A=-1 \]

\[ \boxed{ y=C_1e^x+C_2e^{2x}-xe^x } \]

GATE Problem Bank - Partial Differential Equations (PDE)

Problem 11: Form PDE by Eliminating Constants

Question:

Given: \[ z=ax^2+by^2 \] Form the PDE by eliminating constants \(a,b\).

Step 1

Differentiate with respect to x: \[ \frac{\partial z}{\partial x}=2ax \] Therefore: \[ a=\frac{z_x}{2x} \] Differentiate with respect to y: \[ \frac{\partial z}{\partial y}=2by \] Therefore: \[ b=\frac{z_y}{2y} \]

Step 2

Substitute into original equation: \[ z= \frac{xz_x}{2} + \frac{yz_y}{2} \] Multiply by 2:

\[ \boxed{ 2z=xz_x+yz_y } \]

Problem 12: Form PDE by Eliminating Arbitrary Function

Question:

\[ z=f(x^2+y^2) \] Find PDE.

Step 1

Differentiate: \[ z_x=f'(x^2+y^2)2x \] \[ z_y=f'(x^2+y^2)2y \]

Step 2

Divide: \[ \frac{z_x}{2x} = \frac{z_y}{2y} \] Therefore:

\[ \boxed{ yz_x-xz_y=0 } \]

Problem 13: Lagrange First Order PDE

Question:

\[ x p+y q=z \] where \[ p=\frac{\partial z}{\partial x}, \quad q=\frac{\partial z}{\partial y} \]

Step 1

Lagrange form: \[ \frac{dx}{x} = \frac{dy}{y} = \frac{dz}{z} \]

Step 2

From first two: \[ \frac{dx}{x}=\frac{dy}{y} \] Integrate: \[ \ln x-\ln y=C \] Therefore: \[ \frac{x}{y}=C_1 \] From: \[ \frac{dx}{x}=\frac{dz}{z} \] \[ \ln x-\ln z=C \] Therefore: \[ \frac{x}{z}=C_2 \]

General solution: \[ \boxed{ F\left(\frac{x}{y},\frac{x}{z}\right)=0 } \]

Problem 14: PDE CF Only

Question:

\[ (D^2-4DD'+3D'^2)z=0 \]

Step 1: Auxiliary equation

\[ m^2-4m+3=0 \] \[ (m-1)(m-3)=0 \] Roots: \[ m=1,3 \]

Step 2: Complementary Function

\[ \boxed{ z=\phi_1(y+x)+\phi_2(y+3x) } \]

Problem 15: PDE CF + PI

Question:

\[ (D^2+3DD'+2D'^2)z=e^{x+2y} \]

Step 1: CF

Auxiliary equation: \[ m^2+3m+2=0 \] \[ (m+1)(m+2)=0 \] Therefore: \[ CF=\phi_1(y-x)+\phi_2(y-2x) \]

Step 2: PI

For exponential: \[ D=1,\quad D'=2 \] Substitute: \[ F(1,2)=1+6+8 \] \[ =15 \] Therefore: \[ PI=\frac1{15}e^{x+2y} \]

\[ \boxed{ z= \phi_1(y-x)+ \phi_2(y-2x) +\frac1{15}e^{x+2y} } \]

Problem 16: PDE Exponential Trial Function

Question:

Find \(\xi,\eta\) such that: \[ f_{xx}+2f_{xy}+f_{yy}=4f \] Assume: \[ f=e^{\xi x+\eta y} \]

Step 1

Convert derivatives: \[ f_{xx}=\xi^2f \] \[ f_{xy}=\xi\eta f \] \[ f_{yy}=\eta^2f \]

Step 2

Substitute: \[ (\xi^2+2\xi\eta+\eta^2)f=4f \] Cancel \(f\):

\[ \boxed{ (\xi+\eta)^2=4 } \]

Problem 17: Second Order PDE Classification

Question:

Classify: \[ 3u_{xx}+4u_{xy}+u_{yy}=0 \] Compare: \[ Au_{xx}+Bu_{xy}+Cu_{yy}=0 \] Here: \[ A=3,\quad B=4,\quad C=1 \] Calculate: \[ B^2-4AC \] \[ =16-12 \] \[ =4 \]

Since: \[ B^2-4AC>0 \] \[ \boxed{\text{Hyperbolic PDE}} \]

Problem 18: Heat Equation Separation

Question:

\[ u_t=\alpha^2u_{xx} \] Assume: \[ u=X(x)T(t) \] Substitute: \[ XT'=\alpha^2X''T \] Separate: \[ \frac{T'}{\alpha^2T} = \frac{X''}{X} = -\lambda^2 \] Solutions: \[ T=e^{-\alpha^2\lambda^2t} \] \[ X=A\sin\lambda x+B\cos\lambda x \]

\[ \boxed{ u= (A\sin\lambda x+B\cos\lambda x) e^{-\alpha^2\lambda^2t} } \]

Problem 19: PDE PI Failure Case

Question:

\[ (D-D')z=e^{x+y} \] For PI: \[ D=1,D'=1 \] Denominator: \[ D-D'=0 \] Failure case.

Correction

Multiply by x: \[ PI=xe^{x+y} \]

\[ \boxed{ z=\phi(x+y)+xe^{x+y} } \]

Problem 20: Laplace Equation

Question:

Solve: \[ u_{xx}+u_{yy}=0 \] Assume: \[ u=X(x)Y(y) \] Substitute: \[ X''Y+XY''=0 \] Divide: \[ \frac{X''}{X} = -\frac{Y''}{Y} \] Set: \[ =-\lambda^2 \] Therefore: \[ X''+\lambda^2X=0 \] \[ Y''-\lambda^2Y=0 \]

General form: \[ \boxed{ u=(A\sin\lambda x+B\cos\lambda x) (Ce^{\lambda y}+De^{-\lambda y}) } \]

GATE Problem Bank - Fourier Series and Laplace Transform

Problem 21: Fourier Series of a Constant Function

Question:

Find Fourier series of: \[ f(x)=1,\quad -\piStep 1: Symmetry Constant function is even. Therefore: \[ b_n=0 \]

Step 2: Calculate \(a_0\)

\[ a_0= \frac1\pi \int_{-\pi}^{\pi}1dx \] \[ a_0=2 \] Therefore: \[ \frac{a_0}{2}=1 \]

Step 3

All cosine coefficients are zero.

\[ \boxed{ f(x)=1 } \]

Problem 22: Fourier Series of \(f(x)=x\)

Question:

Find Fourier series: \[ -\piStep 1: Symmetry Since: \[ f(-x)=-f(x) \] Function is odd. Therefore: \[ a_0=0,\quad a_n=0 \] Only sine terms exist.

Step 2

\[ b_n= \frac2\pi \int_0^\pi x\sin(nx)dx \] Integration: \[ b_n= \frac{2(-1)^{n+1}}n \]

\[ \boxed{ x= 2\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}n \sin nx } \]

Problem 23: Even Function Fourier Series

Question:

Find Fourier series of: \[ f(x)=x^2 \] for: \[ -\piStep 1 \(x^2\) is even. Hence: \[ b_n=0 \]

Step 2

Coefficient: \[ a_n= \frac2\pi \int_0^\pi x^2\cos(nx)dx \] Result: \[ a_n= \frac{4(-1)^n}{n^2} \] Also: \[ a_0=\frac{2\pi^2}{3} \]

\[ \boxed{ x^2= \frac{\pi^2}{3} + 4\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}\cos nx } \]

Problem 24: Half Range Sine Series

Question:

Expand: \[ f(x)=x \] in sine series: \[ 0 \[ \boxed{ x= 2 \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}n \sin nx } \]