Key takeaways
- Eight units, from order and degree to the heat and wave equations.
- Solved examples for variable separable, linear, exact and higher order equations.
- Clear CF and PI rules for constant-coefficient ODEs.
- Ten practice questions at the end in university and GATE style.
- Original Uncodemy revision notes; check your own university syllabus.
About these handwritten notes
These are original Uncodemy revision notes for Differential Equations, written in a clean handwritten style so they feel like a topper's notebook rather than a textbook. They are short on purpose: each unit gives the key definitions, formulas, a solved example or two and the points that are easy to forget.
The notes follow the common syllabus that most Indian B.Tech universities use for this subject, including universities that follow the AICTE model curriculum. They are not official notes from any university. Unit names, order and depth differ from one university to another, so check your own university syllabus and mark any topic that is missing before you rely on the notes for an exam. They focus on methods: which form an equation is in, which method to use and the steps that carry marks.
What's inside the differential equations notes
The notes have eight units. One line on each:
Unit 1: Basics
Order, degree, linear and non-linear equations, and general and particular solutions.
Unit 2: First order, first degree ODEs
Variable separable, homogeneous, linear (integrating factor), Bernoulli and exact equations, with solved examples, plus first order equations of higher degree solvable for p = dy/dx.
Unit 3: Applications of first order ODEs
Growth and decay with half-life, Newton's law of cooling, the RL circuit and orthogonal trajectories.
Unit 4: Higher order linear ODEs with constant coefficients
The complementary function for real, repeated and complex roots, particular integral rules PI = f(x)/F(D), solved examples and applications of second order ODEs.
Unit 5: Cauchy-Euler equations and variation of parameters
Changing a Cauchy-Euler equation to constant coefficients, and the variation of parameters method for any right-hand side.
Unit 6: Series solutions and special functions
Power series near an ordinary point, the Frobenius method, Legendre polynomials with Rodrigues' formula and orthogonality, and the Bessel equation.
Unit 7: Partial differential equations
Lagrange's linear PDE, classification of second order PDEs by B² - 4AC (elliptic, parabolic, hyperbolic) and the separation of variables method for heat and wave problems.
Unit 8: Most asked questions (practice)
Ten mixed practice questions to test every unit.
How to revise differential equations with these notes
- Identify the type first. Before solving, decide: separable, linear, exact, Bernoulli, constant coefficient or Cauchy-Euler. The notes suggest checking first whether the variables can be separated, because it is the quickest route.
- Make a PI rule sheet. Write the particular integral rules for e^(ax), sin ax and cos ax, polynomials and e^(ax) V(x) on one page, including the failure case. Revise it daily before the exam.
- Practise each CF case. Real distinct roots, repeated roots and complex roots each give a different form. Do one example of each.
- Keep variation of parameters neat. Write y1, y2, the Wronskian W and then the two integrals in order. Examiners give step marks.
- For PDEs, follow the sign rule. The notes advise choosing the negative separation constant (-k²) for heat and wave problems, because it gives the decaying or oscillating solutions that match the physics.
- Check numerically. If a question gives an initial value, you can check your solution at a point with the Runge-Kutta (RK4) calculator.
Exam focus: high-weightage topics
The notes end with a tip: GATE often asks first order linear and exact ODEs, the CF of constant-coefficient ODEs, Cauchy-Euler equations and PDE classification. University papers usually include PI rules and variation of parameters.
Topics worth extra time:
- Exactness: check whether an equation is exact and solve it, or find an integrating factor.
- Linear and Bernoulli equations: dy/dx + Py = Q and its Bernoulli form.
- Higher order ODEs: full solution with CF and PI, including cases like x e^(-x) and sin 2x on the right side.
- Variation of parameters: classic examples such as y'' + y = sec x.
- Cauchy-Euler: equations of the form x² y'' + x y' + y = f(x).
- Series solution near x = 0 and Legendre polynomial properties.
- PDEs: classify a second order PDE and solve the one-dimensional heat equation with given end conditions.
- Applications: a Newton's law of cooling numerical.
Typical questions
The practice set has questions like "Solve dy/dx + y tan x = sec x", "Solve y'' + 4y = sin 2x" and "Solve y'' + y = sec x by variation of parameters". These are practice questions, not a guarantee of what your paper will ask.
Who these notes are for
- First and second-year B.Tech students of any branch. Depending on the university, ODEs sit in Engineering Mathematics 1, 2 or 3.
- GATE aspirants revising differential equations for the engineering mathematics section.
- Students of core subjects such as circuits, control systems, heat transfer and vibrations, where the same ODE methods appear again.
For more unit-wise practice, the Engineering Mathematics 2 important questions PDF has higher order ODE questions with answer outlines. If your course solves ODEs with Laplace transforms, see the Laplace and Fourier transforms notes as well.
Common mistakes to avoid in differential equations
- Writing the linear form wrongly. Before finding the integrating factor, bring the equation to dy/dx + Py = Q. A missing sign in P changes the whole answer.
- Not checking exactness properly. Compare ∂M/∂y and ∂N/∂x carefully. If they are not equal, look for an integrating factor instead of forcing the exact method.
- Ignoring the failure case of a PI rule. When F(a) = 0, the standard rule fails and you need the extra x factor. Write this case on your rule sheet.
- Leaving out the CF. The general solution is CF + PI. Many answers lose marks because only the PI is written.
- Forgetting the constants in initial value problems. Apply the given conditions only after writing the full general solution.
- Wrong separation constant in PDEs. Follow the notes: use -k² for heat and wave equations so the solution decays or oscillates.
A simple 5-day revision plan
- Day 1: Units 1 and 2. All first order methods.
- Day 2: Unit 3 and the CF part of Unit 4.
- Day 3: PI rules from Unit 4, then Unit 5.
- Day 4: Units 6 and 7. Series solutions, Legendre and Bessel, PDEs.
- Day 5: Solve the ten practice questions with a timer and revise your PI rule sheet.
Download Differential Equations Handwritten Notes PDF
Differential Equations Handwritten Notes PDF
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