## Differential Equations

# Formation of a Differential Equation whose General Solution is given

- The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding the family of curves.
- To form a differential equation from a given function we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants.

### Part1: View the Topic in this video From 00:13 To 12:28

### Part2: View the Topic in this video From 00:12 To 12:50

### Part3: View the Topic in this video From 00:12 To 07:02

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- The differential equation of y = Ae
^{αx}+ Be^{βx}is \frac{d^{2}y}{dx^{2}}-(\alpha+\beta)\frac{dy}{dx}+\alpha\beta y = 0 - The differential equation of y = (A + Bx) e
^{kx}is \frac{d^{2}y}{dx^{2}}-2k\frac{dy}{dx}+k^{2} y = 0 - The differential equation of y = A cos kx + B sin kx by eliminating A and B is \frac{d^{2}y}{dx^{2}}+k^{2} y = 0
- The differential equation of y = Ae
^{kx}+ Be^{-kx}by eliminating A and B is \frac{d^{2}y}{dx^{2}}-k^{2} y = 0 - The differential equation of y = e
^{x}(A cos nx + B sin nx) by eliminating A and B is y_{2}− 2y_{1}+ (n^{2}+ 1) y = 0 - The differential equation of y = e
^{kx}(A + Bx + Cx^{2}) by eliminating A, B, C is y_{3}− 3ky_{2}+ 3k^{2}y_{1}− k^{3}y = 0 - The differential equation of y = e
^{kx}(A cos (lx)) + (B sin (lx)) by eliminating A, B is y_{2}− 2ky_{1}+ (k^{2}+ l^{2}) y = 0