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Differential Equations revision notes

Every idea in the Differential Equations chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 21 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Order, degree and formation

Order and degree
1× asked, last 2021
order=highest derivative present,degree=power of that derivative\text{order}=\text{highest derivative present},\qquad \text{degree}=\text{power of that derivative}
Degree exists only when the DE is a polynomial in y′,y′′,…y', y'', \dots after clearing radicals and fractional powers.
  • Not a degree for y′′+sin⁡y′=0y''+\sin y'=0. A derivative inside sin, exp or log is no polynomial term. Rearrange first: ey′=xe^{y'}=x is y′=ln⁡xy'=\ln x, degree 1. Clear roots: y=xy′+1+y′2y=xy'+\sqrt{1+y'^2} has degree 2.
Forming a DE from a family of curves
6× asked, last 2024
n arbitrary constants ⇒ differentiate n times and eliminate ⇒ order nn\ \text{arbitrary constants}\ \Rightarrow\ \text{differentiate } n \text{ times and eliminate}\ \Rightarrow\ \text{order } n
Count independent constants only: y=aex+b=Aexy=ae^{x+b}=Ae^x has one. A given size (radius, latus rectum) is not a constant.
xxyyb>0b>0b<0b<0
x2=4b(y+b)x^2=4b(y+b) for several bb (all with focus OO, red): one constant, so one differentiation gives x(y′)2=x+2yy′x(y')^2=x+2yy'
  • Not a second-order answer for a one-parameter family. x2=4b(y+b)x^2=4b(y+b) has only bb: differentiate once, put b=x/(2y′)b=x/(2y') back, get x(y′)2=x+2yy′x(y')^2=x+2yy'. Two constants, as in y=c1f(x)+c2y=c_1f(x)+c_2, give order 2.

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Practise Differential Equations

19 more concepts in 4 topics

Variable separable and reducible forms, Homogeneous equations, Linear equations and Bernoulli form and Initial values and applications. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Variable separable and reducible forms

  • Variable separable28×
  • Substitution t = ax + by + c8×
  • Exact differentials to spot17×

Homogeneous equations

  • Homogeneous equation: put y = vx16×
  • Homogeneous in x/y: put x = vy4×
  • Shift the origin: (ax+by+c)/(px+qy+r)3×
  • Power substitutions: t = y², x = vy²6×

Linear equations and Bernoulli form

  • Linear equation in y40×
  • Integrating factors to know24×
  • Constant coefficients: y' = ky + c10×
  • Linear in x: dx/dy + P(y)x = Q(y)9×
  • Substitution t = f(y) to make it linear12×
  • Bernoulli equation5×

Initial values and applications

  • Initial-value problems23×
  • Using the solution: max, limit, area25×
  • Growth, decay and cooling5×
  • Curves from a slope condition11×
  • Integral equations that become DEs12×
  • Second-order forms4×

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Practise Differential Equations

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