First Order Linear DE

• Standard Form: \(\dfrac{dy}{dx} + P(x)y = Q(x)\)

• Solution Method:

  1. Find integrating factor: \(\mu(x) = e^{\int P(x)dx}\)

  2. Multiply equation by \(\mu(x)\)

  3. Left side becomes \(\dfrac{d}{dx}[\mu(x)y]\)

  4. Integrate: \(\mu(x)y = \int \mu(x)Q(x)dx + C\)

• General Solution:

  \[y = \dfrac{1}{\mu(x)}\left[\int \mu(x)Q(x)dx + C\right]\]

First Order Nonlinear DE - Separable

• Form: \(\dfrac{dy}{dx} = f(x)g(y)\)

• Method:

  1. Separate variables: \(\dfrac{dy}{g(y)} = f(x)dx\)

  2. Integrate both sides: \(\int \dfrac{dy}{g(y)} = \int f(x)dx + C\)

• Example: \(\dfrac{dy}{dx} = xy\)

  \[\begin{aligned} \dfrac{dy}{y} &= x dx\\ \ln|y| &= \dfrac{x^2}{2} + C\\ y &= Ae^{x^2/2} \end{aligned}\]

Bernoulli’s Equation

• Form: \(\dfrac{dy}{dx} + P(x)y = Q(x)y^n\) where \(n \neq 0, 1\)

• Method:

  1. Divide by \(y^n\): \(y^{-n}\dfrac{dy}{dx} + P(x)y^{1-n} = Q(x)\)

  2. Substitute \(v = y^{1-n}\), so \(\dfrac{dv}{dx} = (1-n)y^{-n}\dfrac{dy}{dx}\)

  3. Get linear equation in \(v\): \(\dfrac{dv}{dx} + (1-n)P(x)v = (1-n)Q(x)\)

  4. Solve for \(v\), then find \(y = v^{1/(1-n)}\)

Exact Differential Equations

• Form: \(M(x,y)dx + N(x,y)dy = 0\)

• Exactness Condition: \(\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}\)

• Solution Method:

  1. Check exactness condition

  2. Find \(F(x,y)\) such that \(\dfrac{\partial F}{\partial x} = M\) and \(\dfrac{\partial F}{\partial y} = N\)

  3. \(F(x,y) = \int M dx + g(y)\) where \(g'(y) = N - \dfrac{\partial}{\partial y}\int M dx\)

  4. Solution: \(F(x,y) = C\)

• If not exact: Find integrating factor \(\mu(x)\) or \(\mu(y)\)

Homogeneous Linear DE

Form: \(a_ny^{(n)} + a_{n-1}y^{(n-1)} + \cdots + a_1y' + a_0y = 0\)

Characteristic Equation: \(a_nr^n + a_{n-1}r^{n-1} + \cdots + a_1r + a_0 = 0\)

Solution based on roots:

• Distinct real roots \(r_1, r_2, \ldots\): \(y = c_1e^{r_1x} + c_2e^{r_2x} + \cdots\)

• Repeated real root \(r\) (multiplicity \(m\)): \(y = (c_1 + c_2x + \cdots + c_mx^{m-1})e^{rx}\)

• Complex roots \(\alpha \pm \beta i\): \(y = e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))\)

Non-homogeneous Linear DE

Form: \(a_ny^{(n)} + a_{n-1}y^{(n-1)} + \cdots + a_1y' + a_0y = f(x)\)

General Solution: \(y = y_h + y_p\)

• \(y_h\): homogeneous solution

• \(y_p\): particular solution

Method of Undetermined Coefficients: For \(f(x) =\) polynomial, exponential, sine, cosine, or their products:

• \(f(x) = P_n(x)\) \(\Rightarrow\) \(y_p = x^s Q_n(x)\)

• \(f(x) = e^{ax}\) \(\Rightarrow\) \(y_p = x^s Ae^{ax}\)

• \(f(x) = \sin(bx), \cos(bx)\) \(\Rightarrow\) \(y_p = x^s(A\cos(bx) + B\sin(bx))\)

where \(s\) = multiplicity of root in characteristic equation

Variation of Parameters

For \(y'' + p(x)y' + q(x)y = f(x)\) with homogeneous solutions \(y_1, y_2\):

Particular Solution: \(y_p = u_1y_1 + u_2y_2\)

where:

Wronskian: \(W(y_1,y_2) = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1y_2' - y_1'y_2\)

Steps:

1. Find \(y_1, y_2\) from homogeneous equation

2. Calculate Wronskian \(W(y_1,y_2)\)

3. Find \(u_1' = -\dfrac{y_2f(x)}{W}\), \(u_2' = \dfrac{y_1f(x)}{W}\)

4. Integrate to get \(u_1, u_2\)

5. \(y_p = u_1y_1 + u_2y_2\)

Cauchy-Euler Equation

Form: \(x^ny^{(n)} + a_{n-1}x^{n-1}y^{(n-1)} + \cdots + a_1xy' + a_0y = 0\)

Method: Substitute \(x = e^t\) or \(y = x^r\)

For second order: \(x^2y'' + bxy' + cy = 0\)

1. Assume \(y = x^r\)

2. Get characteristic equation: \(r(r-1) + br + c = 0\)

3. Solve: \(r^2 + (b-1)r + c = 0\)

Solutions:

• Distinct real roots: \(y = c_1x^{r_1} + c_2x^{r_2}\)

• Repeated root: \(y = (c_1 + c_2\ln x)x^r\)

• Complex roots \(\alpha \pm \beta i\): \(y = x^\alpha(c_1\cos(\beta\ln x) + c_2\sin(\beta\ln x))\)

Initial Value Problems (IVP)

Form: DE + Initial conditions at single point

Example:

Method:

1. Solve homogeneous DE: \(y = c_1e^{-x} + c_2e^{-3x}\)

2. Apply initial conditions:

   \[\begin{aligned} y(0) = c_1 + c_2 &= 1\\ y'(0) = -c_1 - 3c_2 &= 2 \end{aligned}\]

3. Solve for constants: \(c_1 = \dfrac{5}{2}, c_2 = -\dfrac{3}{2}\)

4. Solution: \(y = \dfrac{5}{2}e^{-x} - \dfrac{3}{2}e^{-3x}\)

Boundary Value Problems (BVP)

Form: DE + Boundary conditions at different points

Example:

Eigenvalue Problem:

• For \(\lambda > 0\): \(y = A\cos(\sqrt{\lambda}x) + B\sin(\sqrt{\lambda}x)\)

• From \(y(0) = 0\): \(A = 0\)

• From \(y(L) = 0\): \(B\sin(\sqrt{\lambda}L) = 0\)

• Non-trivial solution: \(\sin(\sqrt{\lambda}L) = 0\)

• Eigenvalues: \(\lambda_n = \dfrac{n^2\pi^2}{L^2}\), \(n = 1,2,3,\ldots\)

• Eigenfunctions: \(y_n = \sin\left(\dfrac{n\pi x}{L}\right)\)

Classification of PDEs

General Second Order PDE:

Classification:

• Elliptic: \(B^2 - 4AC < 0\) (e.g., Laplace equation)

• Parabolic: \(B^2 - 4AC = 0\) (e.g., Heat equation)

• Hyperbolic: \(B^2 - 4AC > 0\) (e.g., Wave equation)

Important PDEs:

• Heat Equation: \(u_t = \alpha^2 u_{xx}\)

• Wave Equation: \(u_{tt} = c^2 u_{xx}\)

• Laplace Equation: \(u_{xx} + u_{yy} = 0\)

Method of Separation of Variables

Technique: Assume \(u(x,t) = X(x)T(t)\)

Example - Heat Equation: \(u_t = \alpha^2 u_{xx}\)

1. Substitute: \(X(x)T'(t) = \alpha^2 X''(x)T(t)\)

2. Separate: \(\dfrac{T'(t)}{\alpha^2 T(t)} = \dfrac{X''(x)}{X(x)} = -\lambda\)

3. Get ODEs:

   \[\begin{aligned} T'(t) + \alpha^2\lambda T(t) &= 0\\ X''(x) + \lambda X(x) &= 0 \end{aligned}\]

4. Solve each ODE

5. Apply boundary/initial conditions

6. General solution: \(u(x,t) = \sum_{n=1}^{\infty} A_n X_n(x) T_n(t)\)

Heat Equation - Complete Solution

Problem: \(u_t = \alpha^2 u_{xx}\), \(0 < x < L\), \(t > 0\)

Boundary Conditions: \(u(0,t) = u(L,t) = 0\)

Initial Condition: \(u(x,0) = f(x)\)

Solution Steps:

1. Separation gives eigenvalue problem: \(X''+ \lambda X = 0\), \(X(0) = X(L) = 0\)

2. Eigenvalues: \(\lambda_n = \dfrac{n^2\pi^2}{L^2}\), \(X_n = \sin\dfrac{n\pi x}{L}\)

3. Time equation: \(T_n = e^{-\alpha^2\lambda_n t}\)

4. General solution: \(u(x,t) = \sum_{n=1}^{\infty} A_n \sin\dfrac{n\pi x}{L} e^{-\alpha^2 n^2\pi^2 t/L^2}\)

5. Fourier coefficients: \(A_n = \dfrac{2}{L}\int_0^L f(x)\sin\dfrac{n\pi x}{L}dx\)

Wave Equation Solution

Problem: \(u_{tt} = c^2 u_{xx}\), \(0 < x < L\), \(t > 0\)

Boundary Conditions: \(u(0,t) = u(L,t) = 0\)

Initial Conditions: \(u(x,0) = f(x)\), \(u_t(x,0) = g(x)\)

Solution:

where:

Key Formulas - Quick Reference

Linear First Order: \(y = \dfrac{1}{\mu(x)}\left[\int \mu(x)Q(x)dx + C\right]\), \(\mu = e^{\int P dx}\)

Characteristic Equation Roots:

• Real distinct: \(c_1e^{r_1x} + c_2e^{r_2x}\)

• Real repeated: \((c_1 + c_2x)e^{rx}\)

• Complex \(\alpha \pm \beta i\): \(e^{\alpha x}(c_1\cos\beta x + c_2\sin\beta x)\)

Wronskian: \(W = y_1y_2' - y_1'y_2\)

Variation of Parameters: \(u_1' = -\dfrac{y_2f}{W}\), \(u_2' = \dfrac{y_1f}{W}\)

Fourier Coefficients: \(A_n = \dfrac{2}{L}\int_0^L f(x)\sin\dfrac{n\pi x}{L}dx\)

Important Tips for GATE

• Always check initial/boundary conditions carefully

• For Cauchy-Euler equations, try \(y = x^r\)

• Method of undetermined coefficients works only for specific \(f(x)\)

• Use variation of parameters for general \(f(x)\)

• In separation of variables, the separation constant determines the nature of solution

• Eigenvalue problems often have infinite solutions

• Fourier series representation is key for PDE solutions

• Practice standard boundary conditions: Dirichlet, Neumann, Mixed