Differential Equations

Maths · Class 12

Simulation · Maths · Class 12

A family of solution curves

"A family of solution curves" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "General and particular solutions" in Differential Equations. Each short mark shows the slope the equation demands at that point. Every value of C gives a curve that follows the marks: together they are the general solution. Try both equations. A first-order equation has a one-constant family of solutions. One condition, such as a point the curve must pass through, fixes the constant. JEE Main asked 57 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).

From the lesson General and particular solutions in Differential Equations. Change the values and watch what happens.

A family of solution curvesMaths · Class 12

In JEE Main

42 shifts analysed

57 questions from Differential Equations across the 42 JEE Main shifts Lumi has analysed (2017–2026).

See the past questions

The idea behind it

NCERT §9.3

  • A solution of a differential equation is a function that makes both sides equal when it and its derivatives are substituted. Its graph is called a solution curve (or integral curve).
  • Worked example: y = e^(2x) solves y″ − 3y′ + 2y = 0, because 4e^(2x) − 6e^(2x) + 2e^(2x) = 0. So does y = eˣ: 1 − 3 + 2 = 0.
  • A solution that contains arbitrary constants is the general solution. Giving the constants particular values produces a particular solution.
  • Worked example: y = A cos 3x + B sin 3x solves y″ + 9y = 0 for every A and B. The equation has order 2 and the general solution has two constants.
  • Conditions fix the constants. With y(0) = 2 and y′(0) = 6: A = 2 and 3B = 6, so B = 2 and the particular solution is y = 2 cos 3x + 2 sin 3x.
  • Worked example: the general solution of dy/dx = 2x is y = x² + C, a family of parabolas stacked vertically. The one through (1, 4) has 4 = 1 + C, so C = 3 and y = x² + 3.
  • Picture a first-order equation as a field of short slope marks: at each point (x, y) it gives the slope dy/dx. A solution curve follows the marks, and a starting point picks out one curve of the family.

Questions

3 answers

What does the "A family of solution curves" simulation show?
Each short mark shows the slope the equation demands at that point. Every value of C gives a curve that follows the marks: together they are the general solution. Try both equations.
Does JEE Main ask questions from Differential Equations?
Yes. JEE Main asked 57 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
Is this simulation free?
Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "General and particular solutions" in Differential Equations (Class 12 Maths).

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