Questions
3 questions per exam
Difficulty
Medium
Importance
High yield for HPCL/NTPC/ONGC
Overview
Differential Equations serve as the mathematical backbone for modeling physical systems in engineering, ranging from heat transfer to wave mechanics. Mastery of this topic is essential for competitive PSU exams as it accounts for recurring analytical and application-based questions that test both conceptual clarity and calculation speed.
First Order Ordinary Differential Equations
First order equations describe rate-of-change relationships and are categorized by their solution techniques. In PSU exams, focus on identifying the correct form to apply the appropriate integrating factor or variable separation method.
- Variable Separable: dy/dx = f(x)g(y)
- Linear Form: dy/dx + Py = Q; IF = e^∫Pdx
- Bernoulli Equation: dy/dx + Py = Qy^n; substitute v = y^(1-n)
- Exact Equation condition: Mdx + Ndy = 0 if ∂M/∂y = ∂N/∂x
- Clairaut's Equation: y = px + f(p); solution is y = cx + f(c)
Higher Order Linear Differential Equations
These equations involve derivatives higher than the first order and are typically solved using the Auxiliary Equation (AE). Success depends on correctly determining the Complementary Function (CF) and the Particular Integral (PI).
- AE: f(D)y = 0; roots m1, m2 determine CF structure
- Distinct roots: y = C1e^(m1x) + C2e^(m2x)
- Repeated 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
Partial Differential Equations (PDEs)
PDEs deal with multiple independent variables and are fundamental to field theory and thermodynamics. The method of separation of variables is the primary tool for solving BVP in engineering applications.
- Heat Equation: ∂u/∂t = α(∂^2u/∂x^2)
- Wave Equation: ∂^2u/∂t^2 = c^2(∂^2u/∂x^2)
- Laplace Equation: ∂^2u/∂x^2 + ∂^2u/∂y^2 = 0
- Separation of Variables: Assume u(x,t) = X(x)T(t)
- Linearity: Superposition principle applies to homogeneous PDEs
Formula Sheet
Integrating Factor (IF) = e^∫Pdx
General Solution = CF + PI
Cauchy-Euler Eq: x^2(d^2y/dx^2) + ax(dy/dx) + by = 0
D'Alembert's solution for Wave Equation: u(x,t) = 0.5[f(x-ct) + f(x+ct)]
PI for polynomial: Use binomial expansion of [f(D)]^-1
Condition for Exactness: ∂M/∂y = ∂N/∂x
Exam Tip
Always verify if the denominator in the Particular Integral formula is zero; if it is, multiply by x and differentiate the denominator before re-evaluating.
Common Mistakes
- Forgetting to check the condition ∂M/∂y = ∂N/∂x before solving as an exact differential equation
- Miscalculating the Particular Integral when the denominator becomes zero after substituting the value of 'a'
- Confusing the general solution structure for repeated versus distinct roots in the auxiliary equation
More Revision Notes
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