Questions
2 questions per paper
Difficulty
Medium
Importance
Medium yield for PSU technical sections
Overview
Differential Equations serve as the mathematical backbone for modeling physical systems in engineering. For PSU exams, mastering the identification and solution techniques for first and higher-order equations is essential as they frequently appear in technical papers for disciplines like Electrical and Mechanical engineering.
First Order Ordinary Differential Equations
First-order equations represent the simplest dynamic relationships where the rate of change depends on the current state. Focus on recognizing exact equations and using integrating factors for linear forms.
- Variable Separable: dy/dx = f(x)g(y)
- Linear Form: dy/dx + Py = Q where P and Q are functions of x
- Integrating Factor (IF) = e^(integral of P dx)
- Exact Equation condition: dM/dy = dN/dx for Mdx + Ndy = 0
Higher Order Linear ODEs
These equations involve derivatives of higher orders and are primarily solved using the auxiliary equation method. For PSU exams, the focus remains on constant coefficients and finding the Particular Integral (PI) for standard inputs like exponentials and polynomials.
- Auxiliary Equation: f(D)y = 0 gives roots m1, m2
- Case 1: Roots real and distinct: y = C1e^(m1x) + C2e^(m2x)
- Case 2: Roots real and equal: y = (C1 + C2x)e^(mx)
- Particular Integral (PI) for 1/f(D) * e^(ax) = e^(ax)/f(a)
- Euler-Cauchy Equation: Substitute x = e^z to convert to constant coefficients
PDEs — Heat and Diffusion
Partial Differential Equations describe how quantities like temperature distribute over time and space. In technical exams, be familiar with the standard one-dimensional heat equation and its boundary value problems.
- 1D Heat Equation: du/dt = alpha * d^2u/dx^2
- Alpha represents thermal diffusivity
- Steady state condition implies du/dt = 0
- Solution involves separation of variables u(x,t) = X(x)T(t)
- Boundary conditions dictate the harmonic nature of the solution
Formula Sheet
dy/dx + Py = Q -> y * IF = integral(Q * IF) dx
IF = exp(integral(P dx))
PI = 1/f(D) * e^(ax) = e^(ax)/f(a)
1/f(D) * sin(ax) = 1/f(-a^2) * sin(ax)
du/dt = k * d^2u/dx^2
Exam Tip
Always test the boundary conditions in PDE problems first, as they can often eliminate wrong options without needing a full derivation.
Common Mistakes
- Miscalculating the Particular Integral (PI) when the auxiliary equation root equals the exponent constant (a).
- Forgetting to check the 'Exactness' condition before attempting to solve via integration by parts.
- Neglecting the constant of integration in simple variable separable problems, leading to incorrect objective options.
More Revision Notes
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