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 electrical and mechanical engineering. In PSU exams, the focus remains on standard solving techniques for first and higher-order ODEs, which are frequent scorers if one masters the characteristic equation approach. Grasping the distinction between homogeneous and non-homogeneous solutions is essential for quick numerical solving.
First Order ODEs
First order equations describe rate-of-change relationships and are often solved using separation of variables or by identifying them as linear forms. Identifying the correct integrating factor is the most critical step to avoid complex integrations.
- Separable form: dy/dx = f(x)g(y)
- Linear form: dy/dx + P(x)y = Q(x)
- Integrating factor (IF) = e^integral(P dx)
- Solution: y * IF = integral(Q * IF dx) + C
- Exact equation condition: dM/dy = dN/dx
Higher Order Linear ODEs
These equations involve constant coefficients and are solved by finding the auxiliary equation to determine the Complementary Function (CF). The Particular Integral (PI) is then computed based on the nature of the right-hand side function.
- Characteristic/Auxiliary Equation: f(m) = 0
- Real and distinct roots: y = c1e^(m1x) + c2e^(m2x)
- Repeated roots: y = (c1 + c2x)e^(mx)
- Complex roots: y = e^(ax)(c1 cos bx + c2 sin bx)
- PI for e^(ax): 1/f(D) * e^(ax) = e^(ax)/f(a)
Method of Variation of Parameters
This is a versatile technique used to find the particular integral when the RHS does not fit standard operator rules, specifically for non-homogeneous linear equations. It requires knowledge of the Wronskian of the independent solutions.
- General form: y = u*y1 + v*y2
- Wronskian W = y1y2' - y2y1'
- u = -integral(y2 * X / W dx)
- v = integral(y1 * X / W dx)
- Works for any continuous function X(x)
PDEs & Separation of Variables
Partial Differential Equations model multi-dimensional phenomena like heat conduction or wave propagation. The method of separation of variables assumes the solution is a product of single-variable functions to reduce the PDE into multiple ODEs.
- Assumed form: u(x,t) = X(x)T(t)
- Heat equation: du/dt = c^2 * d^2u/dx^2
- Wave equation: d^2u/dt^2 = c^2 * d^2u/dx^2
- Laplace equation: d^2u/dx^2 + d^2u/dy^2 = 0
- Boundary conditions are essential for unique solution constants
Formula Sheet
Linear ODE: dy/dx + Py = Q
IF = exp(integral(P dx))
PI for e^(ax) = e^(ax)/f(a) where f(a) != 0
PI for sin(ax) = sin(ax)/f(-a^2) where f(-a^2) != 0
Wronskian W(y1, y2) = y1y2' - y2y1'
General Solution = CF + PI
Exam Tip
Always evaluate the Complementary Function first; in many PSU exam options, you can eliminate wrong answers simply by verifying the roots of the auxiliary equation.
Common Mistakes
- Forgetting to check if the denominator f(a) is zero when calculating the Particular Integral (PI) for e^(ax).
- Failing to multiply by 'x' when roots of the auxiliary equation are identical, leading to linearly dependent solutions.
- Incorrectly identifying an exact differential equation by failing to check the partial derivative condition.
More Revision Notes
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