Questions
~3 questions per paper
Difficulty
Medium
Importance
High yield for HPCL/NTPC/ONGC
Overview
Differential Equations serve as the mathematical backbone for modeling physical systems in engineering, representing how variables change with respect to one another. For PSU exams, mastering the classification and systematic solution of ODEs and PDEs is essential for solving technical problems in thermodynamics, fluid mechanics, and structural analysis.
First Order ODEs
First-order equations relate the rate of change of a variable to its current state. You must quickly identify the form to choose the correct analytical method, as time is a constraint in exams like HPCL and BHEL.
- Separation of Variables: dy/dx = f(x)g(y)
- Linear Form: dy/dx + Py = Q, Integrating Factor (IF) = e^(integral P dx)
- Bernoulli Equation: dy/dx + Py = Qy^n, solved by substituting v = y^(1-n)
- Exact Equations: M dx + N dy = 0 where partial M/partial y = partial N/partial x
Higher Order Linear ODEs
These equations involve derivatives of order two or higher, typically with constant coefficients. Candidates must be proficient in finding the complementary function (CF) and the particular integral (PI) for various forcing functions.
- Auxiliary Equation: f(D)y = 0 yields roots for CF
- 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 Equation: x^2(d^2y/dx^2) + ax(dy/dx) + by = f(x)
PDEs and Flow Equations
Partial Differential Equations describe systems where more than one independent variable influences the outcome, such as temperature distribution or fluid velocity. Focus on the standard heat and wave equations which are frequent in PSU technical sections.
- 1D Heat Equation: delta(u)/delta(t) = alpha * delta^2(u)/delta(x^2)
- Laplace Equation: del^2(u) = 0 for steady-state systems
- Wave Equation: delta^2(u)/delta(t^2) = c^2 * delta^2(u)/delta(x^2)
- Method of Separation of Variables for boundary conditions
Formula Sheet
Integrating Factor (IF) = e^(integral P dx)
General Solution (Linear): y * IF = integral (Q * IF) dx + C
Roots of Auxiliary Equation (m1, m2) -> CF = C1e^(m1x) + C2e^(m2x)
PI for polynomial x^n: use Binomial Expansion of [f(D)]^-1
Heat Equation: du/dt = alpha(d^2u/dx^2)
Exam Tip
Always verify your solution by substituting the constants back into the original differential equation if you have 30 seconds to spare; it often reveals sign errors.
Common Mistakes
- Miscalculating the Integrating Factor (IF) by forgetting the sign of the P coefficient.
- Failing to account for repeated roots in the Auxiliary Equation, leading to missing the x term in the solution.
- Neglecting the validity condition when PI involves a denominator that evaluates to zero (Case of Failure).
More Revision Notes
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