Question

Verify that the indicated function y = ϕ(x) is an explicit solution of the given first-order differential equation. y' = 2xy2; y = 1/(16 − x2)

Answer 1

Answer with Step-by-step explanation:

We are given that a differential equation

$y'=2xy^2$ and $y=\frac{1}{16-x^2}=(16-x^2)^{-1}$

We have to verify that the function y=$\phi(x)$ is an explicit function of the given first order differential equation.

Differentiate w.r.t x

$y'=-(16-x^2)^{-2}(-2x)$

By using the formula

$\frac{dx^n}{dx}=nx^{n-1}$

$y'=\frac{2x}{(16-x^2)^2}$

Substitute the value of y in given differential equation

$y'=2x\times (\frac{1}{16-x^2})^2$

$y'=\frac{2x}{(16-x^2)^2}$

LHS=RHS

Hence, the function y is an explicit function of the given first order differential equation.