Question

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b: $y = e^{2x} (a + bx)$

Answer 1

Answer:

The required differential equation is $\frac{d^2y}{dx^2}-4\frac{dy}{dx}+4y=0$

Step-by-step explanation:

Concept:

Differential equations are formed by eliminating arbitrary constants occur in the equation.

$y=e^{2x}(a+bx)\\\\e^{-2x}.y=a+bx$

Differentiate with respect to x

$e^{-2x}.\frac{dy}{dx}+y.e^{-2x}(-2)=b\\\\e^{-2x}.[\frac{dy}{dx}-2y]=b$

Once again differentiate with respect to x

$e^{-2x}.[\frac{d^2y}{dx^2}-2\frac{dy}{dx}]+(-2)e^{-2x}.[\frac{dy}{dx}-2y]=0\\\\e^{-2x}.[\frac{d^2y}{dx^2}-2\frac{dy}{dx}-2\frac{dy}{dx}+4y]=0\\\\e^{-2x}.[\frac{d^2y}{dx^2}-4\frac{dy}{dx}+4y]=0$

since$\:e^{-2x}\neq0,$

$\frac{d^2y}{dx^2}-4\frac{dy}{dx}+4y=0$