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 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 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 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: \[ 0Laplace Transform Problem Bank
Problem 25: Basic Laplace Transform
Question:
Find: \[ L\{e^{at}\} \] Formula: \[ L\{e^{at}\} = \int_0^\infty e^{-st}e^{at}dt \] Simplify: \[ = \int_0^\infty e^{-(s-a)t}dt \]
\[
\boxed{
L\{e^{at}\}
=
\frac1{s-a}
}
\]
Problem 26: Laplace of Derivative
Question:
Find: \[ L\{y'\} \] Using integration:
\[
\boxed{
L\{y'\}=sY(s)-y(0)
}
\]
Problem 27: Inverse Laplace
Question:
Find inverse Laplace: \[ Y(s)=\frac1{s(s+1)} \] Partial fractions: \[ \frac1{s(s+1)} = \frac1s-\frac1{s+1} \] Inverse:
\[
\boxed{
y=1-e^{-t}
}
\]
Problem 28: Second Order ODE Using Laplace
Question:
Solve: \[ y''+y=0 \] Conditions: \[ y(0)=0 \] \[ y'(0)=1 \] Laplace: \[ s^2Y-1+Y=0 \] Therefore: \[ Y=\frac1{s^2+1} \]
\[
\boxed{
y=\sin t
}
\]
Problem 29: Unit Step Function
Question:
Find Laplace: \[ u(t-a) \] where \(u\) is step function.
\[
\boxed{
L\{u(t-a)\}
=
\frac{e^{-as}}s
}
\]
Problem 30: Shift Property
Question:
Find: \[ L\{e^{at}f(t)\} \] Using shift:
\[
\boxed{
L\{e^{at}f(t)\}=F(s-a)
}
\]
Problem 31: Convolution Theorem
Question:
If: \[ F(s)=G(s)H(s) \] then:
\[
\boxed{
L^{-1}\{F(s)\}
=
g(t)*h(t)
}
\]
where:
\[
g*h=
\int_0^t g(\tau)h(t-\tau)d\tau
\]
Problem 32: Laplace Transform of Sine
Question:
Find: \[ L\{\sin at\} \]
\[
\boxed{
L\{\sin at\}
=
\frac{a}{s^2+a^2}
}
\]
Problem 33: Laplace Transform of Cosine
Question:
Find: \[ L\{\cos at\} \]
\[
\boxed{
L\{\cos at\}
=
\frac{s}{s^2+a^2}
}
\]
Problem 34: Initial Value Theorem
Question:
Find: \[ y(0) \] if: \[ Y(s)=\frac{5}{s+5} \] Formula: \[ y(0)=\lim_{s\to\infty}sY(s) \] Therefore: \[ = \lim_{s\to\infty} \frac{5s}{s+5} \]
\[
\boxed{
y(0)=5
}
\]
Problem 35: Final Value Theorem
Question:
Find: \[ y(\infty) \] if: \[ Y(s)=\frac1{s(s+2)} \] Formula: \[ y(\infty)= \lim_{s\to0}sY(s) \] Therefore: \[ = \lim_{s\to0} \frac1{s+2} \]
\[
\boxed{
y(\infty)=\frac12
}
\]
GATE Problem Bank - Heat, Wave and Laplace Equations
Problem 36: Classification of Second Order PDE
Question:
Classify: \[ u_{xx}+6u_{xy}+5u_{yy}=0 \] Compare: \[ Au_{xx}+Bu_{xy}+Cu_{yy}=0 \] Therefore: \[ A=1,\quad B=6,\quad C=5 \] Calculate: \[ B^2-4AC \] \[ =36-20 \] \[ =16 \]
Since:
\[
B^2-4AC>0
\]
\[
\boxed{\text{Hyperbolic PDE}}
\]
Problem 37: Classification of Heat Equation
Question:
Classify: \[ u_t=\alpha^2u_{xx} \] Rewrite: \[ \alpha^2u_{xx}-u_t=0 \] Only one spatial second derivative exists.
\[
\boxed{\text{Parabolic PDE}}
\]
Problem 38: Classification of Laplace Equation
Question:
Classify: \[ u_{xx}+u_{yy}=0 \] Here: \[ A=1,B=0,C=1 \] Therefore: \[ B^2-4AC = 0-4 \] \[ =-4 \]
\[
\boxed{\text{Elliptic PDE}}
\]
Problem 39: Separation of Variables for Heat Equation
Question:
Solve: \[ u_t=\alpha^2u_{xx} \] Assume: \[ u=X(x)T(t) \] Substitute: \[ XT'=\alpha^2X''T \] Divide: \[ \frac{T'}{\alpha^2T} = \frac{X''}{X} \] Set: \[ =-\lambda^2 \]Time equation
\[ T'+\alpha^2\lambda^2T=0 \] Solution: \[ T=e^{-\alpha^2\lambda^2t} \]Space equation
\[ X''+\lambda^2X=0 \] Solution: \[ X=A\sin\lambda x+B\cos\lambda x \]
\[
\boxed{
u=(A\sin\lambda x+B\cos\lambda x)
e^{-\alpha^2\lambda^2t}
}
\]
Problem 40: Heat Equation with Boundary Conditions
Question:
\[ u_t=\alpha^2u_{xx} \] Boundary: \[ u(0,t)=0 \] \[ u(L,t)=0 \]Eigenvalue condition
\[ X(0)=0 \] gives: \[ B=0 \] and: \[ X(L)=0 \] gives: \[ \sin(\lambda L)=0 \] Therefore: \[ \lambda_n=\frac{n\pi}{L} \]
Solution:
\[
\boxed{
u=
\sum_{n=1}^{\infty}
A_n
\sin\frac{n\pi x}{L}
e^{-\alpha^2(n\pi/L)^2t}
}
\]
Problem 41: Wave Equation
Question:
Solve: \[ u_{tt}=c^2u_{xx} \] Assume: \[ u=X(x)T(t) \] Substitute: \[ XT''=c^2X''T \] Separate: \[ \frac{T''}{c^2T} = \frac{X''}{X} = -\lambda^2 \] Equations: \[ X''+\lambda^2X=0 \] \[ T''+c^2\lambda^2T=0 \]
\[
\boxed{
u=
(A\sin\lambda x+B\cos\lambda x)
(C\sin c\lambda t+D\cos c\lambda t)
}
\]
Problem 42: Vibrating String Boundary Condition
Question:
String fixed at: \[ x=0,L \] Condition: \[ u(0,t)=u(L,t)=0 \] Eigenvalues: \[ \lambda_n=\frac{n\pi}{L} \]
Allowed modes:
\[
\boxed{
\sin\frac{n\pi x}{L}
}
\]
Problem 43: Laplace Equation Separation
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: \[ X''+\lambda^2X=0 \] \[ Y''-\lambda^2Y=0 \]
\[
\boxed{
u=
(A\sin\lambda x+B\cos\lambda x)
(Ce^{\lambda y}+De^{-\lambda y})
}
\]
Problem 44: Steady State Heat Equation
Question:
Find steady temperature: \[ u_{xx}=0 \] Integrate: \[ u_x=A \] Again: \[ u=Ax+B \] Boundary: \[ u(0)=T_1 \] \[ u(L)=T_2 \] Solving constants:
\[
\boxed{
u=T_1+
\frac{T_2-T_1}{L}x
}
\]
Problem 45: Fourier Solution of Heat Equation
Question:
Initial condition: \[ u(x,0)=f(x) \] Solution: \[ u(x,t) = \sum A_n \sin\frac{n\pi x}{L} e^{-\alpha^2(n\pi/L)^2t} \] where: \[ A_n= \frac2L \int_0^L f(x) \sin\frac{n\pi x}{L}dx \]
\[
\boxed{
A_n
=
\frac2L
\int_0^L
f(x)\sin\frac{n\pi x}{L}dx
}
\]
Problem 46: Wave Equation General Form
Question:
Find solution form: \[ u_{tt}=c^2u_{xx} \]
D'Alembert solution:
\[
\boxed{
u=f(x-ct)+g(x+ct)
}
\]
Problem 47: Separation Constant Meaning
Question:
Why choose: \[ \frac{X''}{X} = \frac{T'}{\alpha^2T} = -\lambda^2 \] Answer: Negative constant produces oscillatory spatial solutions: \[ X''+\lambda^2X=0 \] which satisfy physical boundary conditions.
\[
\boxed{
-\lambda^2
\text{ gives valid eigenfunctions}
}
\]
Problem 48: Boundary Condition Types
| Type | Condition |
|---|---|
| Dirichlet | \[ u(0,t)=0 \] |
| Neumann | \[ u_x(0,t)=0 \] |
| Mixed | \[ u(0,t)=0,\ u_x(L,t)=0 \] |
\[
\boxed{\text{Boundary conditions determine eigenvalues}}
\]
Problem 49: Maximum Principle Idea
Question:
For Laplace equation: \[ \nabla^2u=0 \] The maximum temperature occurs:
\[
\boxed{
\text{On the boundary, not inside the region}
}
\]
Problem 50: Complete GATE PDE Recognition
| Equation | Method |
|---|---|
| \[ F(D,D')z=0 \] | CF |
| \[ F(D,D')z=\phi(x,y) \] | CF + PI |
| \[ u_t=\alpha^2u_{xx} \] | Heat equation |
| \[ u_{tt}=c^2u_{xx} \] | Wave equation |
| \[ u_{xx}+u_{yy}=0 \] | Laplace equation |
\[
\boxed{
\text{Identify the equation before solving}
}
\]
GATE Quick Practice Set (Problems 51-100)
Section A: ODE Practice
Problem 51
Solve:
\[ \frac{dy}{dx}=2x \]
\[
\boxed{y=x^2+C}
\]
Problem 52
\[ y'+y=0 \] Solution: \[ \boxed{y=Ce^{-x}} \]Problem 53
Solve: \[ y''+4y=0 \] Characteristic equation: \[ m^2+4=0 \]
\[
\boxed{
y=C_1\cos2x+C_2\sin2x
}
\]
Problem 54
\[ y''-9y=0 \] Roots: \[ m=\pm3 \]
\[
\boxed{
y=C_1e^{3x}+C_2e^{-3x}
}
\]
Problem 55
\[ (D^2-1)y=0 \]
\[
\boxed{
y=C_1e^x+C_2e^{-x}
}
\]
Problem 56
\[ (D^2+4)y=\sin2x \] RHS repeats in CF.
\[
\boxed{
PI=-\frac{x\cos2x}{4}
}
\]
Problem 57
Cauchy Euler: \[ x^2y''+xy'-y=0 \] Put: \[ y=x^m \] Equation: \[ m^2-1=0 \]
\[
\boxed{
y=C_1x+C_2x^{-1}
}
\]
Problem 58
Laplace: \[ L\{1\} \]
\[
\boxed{\frac1s}
\]
Problem 59
\[ L\{t\} \]
\[
\boxed{\frac1{s^2}}
\]
Problem 60
\[ L\{e^{-at}\} \]
\[
\boxed{
\frac1{s+a}
}
\]
Section B: PDE Practice
Problem 61
Given: \[ f=e^{ax+by} \] Find: \[ f_x \]
\[
\boxed{
f_x=af
}
\]
Problem 62
Find: \[ f_{xx} \] for: \[ f=e^{ax+by} \]
\[
\boxed{
f_{xx}=a^2f
}
\]
Problem 63
Given: \[ af_x+bf_y=f \] Condition:
\[
\boxed{
a\xi+b\eta=1
}
\]
Problem 64
For: \[ f_{xy} \] where: \[ f=e^{\xi x+\eta y} \]
\[
\boxed{
f_{xy}=\xi\eta f
}
\]
Problem 65
PDE: \[ (D-D')z=0 \]
\[
\boxed{
z=\phi(x+y)
}
\]
Problem 66
PDE: \[ (D+D')z=0 \]
\[
\boxed{
z=\phi(y-x)
}
\]
Problem 67
Classification: \[ u_{xx}+u_{yy}=0 \]
\[
\boxed{\text{Elliptic}}
\]
Problem 68
Classification: \[ u_t=u_{xx} \]
\[
\boxed{\text{Parabolic}}
\]
Problem 69
Classification: \[ u_{tt}=u_{xx} \]
\[
\boxed{\text{Hyperbolic}}
\]
Problem 70
Heat equation separation: \[ u=X(x)T(t) \]
\[
\boxed{
T=e^{-\lambda^2t}
}
\]
Section C: Fourier and Laplace Practice
Problem 71
Odd function property:
\[
\boxed{
a_n=0
}
\]
Problem 72
Even function property:
\[
\boxed{
b_n=0
}
\]
Problem 73
Fourier coefficient: \[ a_0 \] represents:
\[
\boxed{
\text{average value}
}
\]
Problem 74
\[ L\{\sin at\} \]
\[
\boxed{
\frac{a}{s^2+a^2}
}
\]
Problem 75
\[ L\{\cos at\} \]
\[
\boxed{
\frac{s}{s^2+a^2}
}
\]
Problem 76
Initial value theorem: \[ y(0)=? \]
\[
\boxed{
\lim_{s\to\infty}sY(s)
}
\]
Problem 77
Final value theorem:
\[
\boxed{
\lim_{s\to0}sY(s)
}
\]
Problem 78
Unit step: \[ u(t-a) \]
\[
\boxed{
\frac{e^{-as}}s
}
\]
Problem 79
Convolution:
\[
\boxed{
L^{-1}(FG)=f*g
}
\]
Problem 80
Laplace derivative:
\[
\boxed{
L(y')=sY-y(0)
}
\]
Section D: Final GATE Concepts
Problems 81-100: Rapid Revision
| Concept | Answer |
|---|---|
| Non-zero RHS PDE | CF + PI |
| Zero RHS PDE | CF only |
| Exponential trial | D → constant |
| Heat equation | Parabolic |
| Wave equation | Hyperbolic |
| Laplace equation | Elliptic |
| Odd Fourier function | Sine terms |
| Even Fourier function | Cosine terms |
| Repeated PI root | Multiply by x |
| Second repeated root ODE | \(xe^{mx}\) |
| Laplace of derivative | \(sY-y(0)\) |
| Laplace of second derivative | \(s^2Y-sy(0)-y'(0)\) |
| Fourier coefficient | Projection integral |
| Boundary conditions | Determine constants/eigenvalues |
| Separation constant | Eigenvalue |
| CF meaning | Homogeneous solution |
| PI meaning | Forced response |
| ODE order | Number of constants |
| PDE order | Highest derivative order |
| GATE strategy | Identify method first |