GATE Differential Equations - One Day Revision Formula Sheet
1. First Order ODE
Standard linear equation:
\[
\frac{dy}{dx}+P(x)y=Q(x)
\]
Integrating factor:
\[
\boxed{
IF=e^{\int P(x)dx}
}
\]
Solution:
\[
\boxed{
y(IF)=\int Q(IF)dx+C
}
\]
2. Exact Differential Equation
Equation:
\[
Mdx+Ndy=0
\]
Condition:
\[
\boxed{
\frac{\partial M}{\partial y}
=
\frac{\partial N}{\partial x}
}
\]
Solution:
\[
\boxed{
\int Mdx+\phi(y)=C
}
\]
3. Second Order Linear ODE
General form:
\[
ay''+by'+cy=0
\]
Auxiliary equation:
\[
\boxed{
am^2+bm+c=0
}
\]
Roots:
| Roots | Solution |
|---|---|
| \(m_1,m_2\) | \[ C_1e^{m_1x}+C_2e^{m_2x} \] |
| Repeated root m | \[ (C_1+C_2x)e^{mx} \] |
| \(\alpha\pm i\beta\) | \[ e^{\alpha x} (C_1\cos\beta x+C_2\sin\beta x) \] |
4. Particular Integral (PI)
For:
\[
F(D)y=X
\]
PI:
\[
\boxed{
\frac1{F(D)}X
}
\]
Important rule:
If RHS is:
\[
e^{ax}
\]
replace:
\[
\boxed{
D\rightarrow a
}
\]
Therefore:
\[
PI=
\frac{e^{ax}}
{F(a)}
\]
5. PI Failure Rules
| Condition | Correction |
|---|---|
| \(F(a)=0\) | Multiply by \(x\) |
| Repeated zero | Multiply by \(x^2\) |
6. Cauchy Euler Equation
Equation:
\[
x^2y''+axy'+by=0
\]
Assume:
\[
\boxed{
y=x^m
}
\]
Auxiliary equation:
\[
m(m-1)+am+b=0
\]
7. PDE Operator Method
Let:
\[
D=\frac{\partial}{\partial x}
\]
\[
D'=\frac{\partial}{\partial y}
\]
PDE:
\[
F(D,D')z=0
\]
Auxiliary equation:
\[
F(m,1)=0
\]
If roots:
\[
m_1,m_2
\]
CF:
\[
\boxed{
z=
\phi_1(y+m_1x)
+
\phi_2(y+m_2x)
}
\]
8. PDE Exponential Trial Method
Assume:
\[
\boxed{
f=e^{\xi x+\eta y}
}
\]
Then:
\[
D\rightarrow\xi
\]
\[
D'\rightarrow\eta
\]
Examples:
\[
f_x=\xi f
\]
\[
f_{xx}=\xi^2f
\]
\[
f_{xy}=\xi\eta f
\]
9. PDE Classification
General PDE:
\[
Au_{xx}+Bu_{xy}+Cu_{yy}=0
\]
Calculate:
\[
\Delta=B^2-4AC
\]
| Condition | Type |
|---|---|
| \(\Delta>0\) | Hyperbolic |
| \(\Delta=0\) | Parabolic |
| \(\Delta<0\) | Elliptic |
10. Heat Equation
Equation:
\[
\boxed{
u_t=\alpha^2u_{xx}
}
\]
Solution:
\[
u=
\sum A_n
\sin\frac{n\pi x}{L}
e^{-\alpha^2(n\pi/L)^2t}
\]
11. Wave Equation
Equation:
\[
\boxed{
u_{tt}=c^2u_{xx}
}
\]
General solution:
\[
\boxed{
u=f(x-ct)+g(x+ct)
}
\]
12. Laplace Equation
Equation:
\[
\boxed{
u_{xx}+u_{yy}=0
}
\]
Used in:
- Electrostatics
- Steady temperature distribution
- Fluid flow
13. Fourier Series
General:
\[
f(x)=
\frac{a_0}{2}
+
\sum
(a_n\cos nx+b_n\sin nx)
\]
Coefficients:
\[
a_n=
\frac2L
\int_{-L}^{L}
f(x)\cos nx\,dx
\]
\[
b_n=
\frac2L
\int_{-L}^{L}
f(x)\sin nx\,dx
\]
14. Fourier Symmetry Shortcut
| Function | Result |
|---|---|
| Odd | \[ a_0=a_n=0 \] Only sine |
| Even | \[ b_n=0 \] Only cosine |
15. Laplace Transform Table
| Function | Transform |
|---|---|
| 1 | \[ \frac1s \] |
| \(t\) | \[ \frac1{s^2} \] |
| \(e^{at}\) | \[ \frac1{s-a} \] |
| \(\sin at\) | \[ \frac{a}{s^2+a^2} \] |
| \(\cos at\) | \[ \frac{s}{s^2+a^2} \] |
16. Laplace Theorems
Derivative:
\[
\boxed{
L(y')=sY-y(0)
}
\]
Second derivative:
\[
\boxed{
L(y'')=
s^2Y-sy(0)-y'(0)
}
\]
Initial value:
\[
\boxed{
y(0)=
\lim_{s\to\infty}sY(s)
}
\]
Final value:
\[
\boxed{
y(\infty)=
\lim_{s\to0}sY(s)
}
\]
17. Final GATE Strategy
| Given | Immediately Think |
|---|---|
| \(F(D)y=0\) | CF |
| \(F(D)y=f(x)\) | CF + PI |
| \(F(D,D')z\) | PDE operator |
| \(e^{\xi x+\eta y}\) | Replace derivatives by constants |
| Boundary + PDE | Separation/Fourier |
| Initial condition | Laplace transform |