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Cheat sheet · JEE Preparation Mathematics

Differential Equations cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

JEE · Mathematics

Cheat sheet

Class 12 · Chapter 9Differential Equations

Linear DE: IF = e^(∫P dx), then y·IF = ∫ Q·IF dx + C.

1. Order & Degree

  • Order: highest derivative.
  • Degree: power of the highest derivative (polynomial form only).
  • General solution has as many constants as the order.

2. Variable Separable

dydx=f(x)g(y)⇒∫dyg(y)=∫f(x)dx\dfrac{dy}{dx} = f(x)g(y) \Rightarrow \int\dfrac{dy}{g(y)} = \int f(x)dx

3. Homogeneous

dydx=F(yx)\dfrac{dy}{dx} = F\left(\dfrac yx\right): put y=vxy = vx, so dydx=v+xdvdx\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}.

4. Linear

dydx+Py=Q\dfrac{dy}{dx} + Py = Q
IF=e∫P dxIF = e^{\int P\,dx}y⋅IF=∫Q⋅IF dx+Cy\cdot IF = \int Q\cdot IF\,dx + C

5. Key Points

  • Particular solution uses initial conditions.
  • Degree undefined if derivative is inside sin, log, etc.
  • Check homogeneity: F(λx, λy) = λⁿF(x, y).
  • x-form: dx/dy + Px = Q also works.

Remember: Order/degree | separable | homogeneous | linear IF

Differential Equations: questions and notes

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