Lesson 1.3: Differential Equations (ODE & PDE, Initial/Boundary Value Problems)

Differential Equations are the foundation of modeling physical systems in Mechanical, Electrical, and Civil Engineering. In GATE and PSU exams, questions are asked on formulation, order, degree, solution methods, and applications.

🔹 1. Basic Definitions

• Ordinary Differential Equation (ODE):
  Involves derivatives with respect to one independent variable.
  Example: dydx+y=0\frac{dy}{dx} + y = 0dxdy​+y=0

• Partial Differential Equation (PDE):
  Involves derivatives with respect to two or more independent variables.
  Example: ∂u∂x+∂u∂y=0\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0∂x∂u​+∂y∂u​=0

• Order of DE: Highest order derivative present.

• Degree of DE: Power of highest order derivative (after removing fractions/roots).

🔹 2. First-Order ODEs

General form:
dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x)dxdy​+P(x)y=Q(x)

Solution (Integrating Factor Method):

y⋅e∫P(x)dx=∫Q(x)e∫P(x)dxdx+Cy \cdot e^{\int P(x) dx} = \int Q(x) e^{\int P(x) dx} dx + Cy⋅e∫P(x)dx=∫Q(x)e∫P(x)dxdx+C

Examples:

1. dydx+y=ex\frac{dy}{dx} + y = e^xdxdy​+y=ex
   👉 Solution: y=(C+e2x2)e−xy = (C + \frac{e^{2x}}{2}) e^{-x}y=(C+2e2x​)e−x

🔹 3. Second-Order Linear ODEs

General form:

ad2ydx2+bdydx+cy=0a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = 0adx2d2y​+bdxdy​+cy=0

Solution Method (Auxiliary Equation):

• Form ar2+br+c=0ar^2 + br + c = 0ar2+br+c=0

• If roots real & distinct → y=C1er1x+C2er2xy = C_1 e^{r_1 x} + C_2 e^{r_2 x}y=C1​er1​x+C2​er2​x

• If roots real & equal → y=(C1+C2x)erxy = (C_1 + C_2 x)e^{rx}y=(C1​+C2​x)erx

• If roots complex → y=eαx(C1cos⁡βx+C2sin⁡βx)y = e^{\alpha x}(C_1 \cos \beta x + C_2 \sin \beta x)y=eαx(C1​cosβx+C2​sinβx)

🔹 4. Partial Differential Equations (PDEs)

• Formulation: Obtained from physical problems like heat conduction, wave equation, fluid flow.

• Standard PDEs:

  • Heat Equation: ∂u∂t=α∂2u∂x2\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}∂t∂u​=α∂x2∂2u​

  • Wave Equation: ∂2u∂t2=c2∂2u∂x2\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}∂t2∂2u​=c2∂x2∂2u​

  • Laplace Equation: ∇2u=0\nabla^2 u = 0∇2u=0

🔹 5. Initial & Boundary Value Problems

• Initial Value Problem (IVP): Value of dependent variable given at starting point.
  Example: y(0)=y0y(0) = y_0y(0)=y0​.

• Boundary Value Problem (BVP): Values given at two boundary points.
  Example: Heat conduction in a rod with fixed end temperatures.

🔹 6. Applications in Engineering

• Mechanical Vibrations → mx¨+cx˙+kx=0m \ddot{x} + c \dot{x} + kx = 0mx¨+cx˙+kx=0

• Heat Transfer → 1D Fourier’s law → PDE

• Fluid Mechanics → Navier–Stokes equation

• Electrical Circuits → Ldidt+Ri+1C∫idt=V(t)L \frac{di}{dt} + Ri + \frac{1}{C}\int i dt = V(t)Ldtdi​+Ri+C1​∫idt=V(t)

🔹 7. Solved Examples (PYQ Style)

Example 1 (GATE 2018 ME):
Solve dydx=y\frac{dy}{dx} = ydxdy​=y, $y(0)=1$.
👉 Solution: y=exy = e^xy=ex.

Example 2 (PSU exam):
Find the general solution of:
d2ydx2−3dydx+2y=0\frac{d^2y}{dx^2} – 3\frac{dy}{dx} + 2y = 0dx2d2y​−3dxdy​+2y=0.
👉 Solution: Auxiliary equation → r2−3r+2=0r^2 – 3r + 2 = 0r2−3r+2=0 → roots (1,2).
So y=C1ex+C2e2xy = C_1 e^x + C_2 e^{2x}y=C1​ex+C2​e2x.

🔹 8. Practice Exercises

1. Solve: dydx+2y=e−x\frac{dy}{dx} + 2y = e^{-x}dxdy​+2y=e−x.

2. Find solution: d2ydx2+9y=0\frac{d^2y}{dx^2} + 9y = 0dx2d2y​+9y=0.

3. Form PDE by eliminating arbitrary constants from $z=ax+by$.

4. Heat conduction in a 1D rod → write governing PDE.

🔹 9. Summary

• ODE: Derivatives w.r.t one variable.

• PDE: Derivatives w.r.t multiple variables.

• Solution techniques: Integrating factor, auxiliary equation, separation of variables.

• IVP & BVP: Used in real-world modeling of thermal, fluid, and vibration systems.