Questions
2 questions per paper
Difficulty
Medium
Importance
Medium yield for HPCL/NTPC/ONGC
Overview
Differential Equations serve as the mathematical foundation for modeling physical systems in engineering, ranging from heat transfer to mechanical vibrations. For PSU exams, mastering the method of finding General and Particular solutions is essential to securing quick marks. The core focus should be on recognizing the standard forms to apply the correct solving technique rapidly.
First Order Ordinary Differential Equations
These equations involve only the first derivative of the dependent variable. Mastery here requires identifying the specific form to apply methods like separation of variables or integrating factors.
- Variable Separable: f(x)dx = g(y)dy
- Linear ODE form: dy/dx + Py = Q
- Integrating Factor (IF): e^(integral P dx)
- Bernoulli Equation: dy/dx + Py = Qy^n
- Exact Equation condition: dM/dy = dN/dx
Higher Order Linear ODEs
Higher order linear equations with constant coefficients are solved by finding the Complementary Function (CF) and the Particular Integral (PI). This is the most frequently tested area in PSU technical exams.
- Auxiliary Equation: f(m) = 0
- PI for e^(ax): 1/f(D) * e^(ax) = e^(ax)/f(a)
- PI for sin(ax) or cos(ax): 1/f(D^2) * sin(ax) = 1/f(-a^2) * sin(ax)
- Cauchy's homogeneous form: Substitute x = e^z
Partial Differential Equations (PDEs)
PSUs focus on the standard boundary value problems in physics, specifically the Heat, Wave, and Laplace equations. Understanding the nature of the coefficients is key for solving these efficiently.
- Heat Equation: du/dt = alpha * 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
- Elliptic (Laplace), Parabolic (Heat), Hyperbolic (Wave) classification
Formula Sheet
dy/dx + Py = Q => y * IF = integral(Q * IF)dx
IF = e^(integral P dx)
Auxiliary Equation: aD^2 + bD + c = 0
Roots of AE: m = [-b +/- sqrt(b^2 - 4ac)] / 2a
PI for x^n: [f(D)]^-1 * x^n (using binomial expansion)
Heat Eq: u(x,t) = (Acos(px) + Bsin(px))e^(-alpha * p^2 * t)
Laplace Eq in 2D: nabla^2 * u = 0
Exam Tip
In the exam, always prioritize finding the Particular Integral first using the shortcut operator formulas to avoid time-consuming standard integration methods.
Common Mistakes
- Forgetting to check the condition for Exact Differential Equations before applying integrating factors.
- Miscalculating the Auxiliary Equation roots, leading to incorrect CF terms (e.g., repeating roots vs real roots).
- Ignoring the condition f(a)=0 when solving for Particular Integral in higher order ODEs.
More Revision Notes
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