Questions
~2 questions per paper
Difficulty
Medium
Importance
Medium yield for GATE and PSU exams
Overview
Differential Equations form the mathematical backbone of engineering analysis, describing how systems evolve over time or space. Mastering these is crucial for PSU exams as they frequently appear in technical papers for mechanical, electrical, and civil engineering roles to model dynamic systems.
First Order ODEs
First-order equations relate the first derivative of a function to the function itself and independent variables. Aspirants should focus on identifying the standard form to apply the correct integrating factor or separation of variables technique.
- Separable: dy/dx = f(x)g(y)
- Linear: dy/dx + Py = Q where P, Q are functions of x
- Integrating Factor (IF) = e^(integral of P dx)
- Bernoulli equation form: dy/dx + Py = Qy^n
- Exact equation condition: dM/dy = dN/dx for Mdx + Ndy = 0
Higher Order Linear ODEs
These equations involve derivatives higher than the first and are essential for solving oscillation and vibration problems. The solution generally consists of a Complementary Function (CF) and a Particular Integral (PI).
- Auxiliary Equation: f(D)y = 0
- PI for e^(ax): 1/f(D) * e^(ax) = 1/f(a) * e^(ax)
- PI for sin(ax) or cos(ax): 1/f(D^2) * sin(ax) = 1/f(-a^2) * sin(ax)
- Euler-Cauchy form: x^n * d^n(y)/dx^n
- Wronskian determinant for linear independence
PDEs — Flow & Diffusion
Partial Differential Equations are used to describe physical phenomena that change with multiple variables like space and time. These are common in thermal and fluid flow applications in exams.
- One-dimensional Heat Equation: du/dt = c^2 * d^2u/dx^2
- One-dimensional Wave Equation: d^2u/dt^2 = c^2 * d^2u/dx^2
- Laplace Equation: d^2u/dx^2 + d^2u/dy^2 = 0
- Method of Separation of Variables
- Steady-state vs Transient analysis
Formula Sheet
y = CF + PI
IF = e^(integral P dx)
y * IF = integral (Q * IF) dx + C
General solution for m1 != m2: y = C1e^(m1x) + C2e^(m2x)
General solution for repeated roots: y = (C1 + C2x)e^(mx)
d^2u/dx^2 + d^2u/dy^2 = 0 (Laplace Equation)
Exam Tip
Always test the 'Auxiliary Equation' roots first for Higher Order ODEs; if roots are repeated, don't forget to multiply by 'x' in your solution.
Common Mistakes
- Forgetting the constant of integration 'C' in indefinite integral solutions for first-order ODEs.
- Neglecting the condition f(a) is not equal to 0 when applying the short-cut formula for Particular Integrals.
- Misidentifying the type of ODE, leading to the use of incorrect integrating factors.
More Revision Notes
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