Questions
~2 questions per paper
Difficulty
Medium
Importance
Medium yield for PSU exams
Overview
Differential Equations serve as the mathematical backbone for modeling physical systems in engineering, ranging from control theory to circuit analysis. Mastering these is essential for PSU exams as they frequently appear in the Engineering Mathematics section, testing both conceptual clarity and calculation speed.
First Order ODEs
First-order equations represent the rate of change of a variable dependent on one independent variable. Success relies on identifying the correct technique, such as Variable Separable or Linear forms, to reach the solution quickly.
- Variable Separable: dy/dx = f(x)g(y)
- Linear Form: dy/dx + Py = Q where P, Q are functions of x
- Integrating Factor (IF): e^(∫P dx)
- Solution: y(IF) = ∫(Q * IF) dx + C
- Bernoulli Equation: dy/dx + Py = Qy^n
Higher Order Linear ODEs
These equations involve derivatives of order two or higher with constant coefficients. The solution is typically the sum of the Complementary Function (CF) and the Particular Integral (PI).
- Characteristic Equation: f(m) = 0
- Distinct roots (m1, m2): y = c1e^(m1x) + c2e^(m2x)
- Equal roots: y = (c1 + c2x)e^(mx)
- PI for e^(ax): 1/f(D) * e^(ax) = 1/f(a) * e^(ax)
- PI for sin(ax) or cos(ax): Replace D^2 with -a^2
Cauchy & Euler Equations
Cauchy-Euler equations are distinguished by variable coefficients where the power of x matches the order of the derivative. They are solved by transforming the variable into a linear equation with constant coefficients.
- Standard Form: x^2(d^2y/dx^2) + ax(dy/dx) + by = 0
- Substitution: x = e^z or z = ln(x)
- Transformation: x(dy/dx) = Dy, x^2(d^2y/dx^2) = D(D-1)y
- Final form is a Linear ODE with constant coefficients in z
PDEs — Separation of Variables
Partial Differential Equations are common in heat conduction and wave motion problems. The separation of variables technique assumes the solution is a product of independent functions of x and t.
- Laplace Equation: ∇^2u = 0
- Heat Equation: ∂u/∂t = α(∂^2u/∂x^2)
- Wave Equation: ∂^2u/∂t^2 = c^2(∂^2u/∂x^2)
- Assumption: u(x,t) = X(x)T(t)
- Constant of separation (k) must be chosen based on boundary conditions
Formula Sheet
y' + Py = Q => y*IF = ∫(Q*IF)dx
IF = e^(∫Pdx)
f(D)y = 0 => Auxiliary Equation: am^n + ... + k = 0
PI = 1/f(D) * X
1/f(D) * e^(ax) = 1/f(a) * e^(ax)
1/(D^2+a^2) * sin(ax) = -x/(2a) * cos(ax)
x^2y'' + axy' + by = 0 => m(m-1) + am + b = 0
Exam Tip
Always verify the form of the Particular Integral (PI) first, as 70% of PSU problems can be solved just by identifying the correct CF/PI structure without fully solving the differential equation.
Common Mistakes
- Forgetting to replace D^2 with -a^2 in the PI calculation for trigonometric terms.
- Miscalculating the Integrating Factor (IF) by omitting the sign of P in the exponent.
- Ignoring the x-coefficient when transforming Cauchy-Euler equations into constant-coefficient ODEs.
More Revision Notes
Ready to test yourself?
Play topic-wise Differential Equations questions in Aspirant Arcade — gamified MCQ practice.
Download Free