Question

what is the separable variable? dy/dx = y² sin x², y(-2) = 1/3?

Answer 1

A differential equation is said to be separable if the variables can be separated. That is, a separable equation is one that can be written in the form. Once this is done, all that is needed to solve the equation is to integrate both sides.

$dy/y^2=sinx^2dx$

$\int dy/y^2=\int sinx^2dx$

$\int dy/y^2=-1/y+c_1$

$\int sinx^2dx$ is is Fresnel integral:

$S(x)=\int sinx^2dx=\displaystyle{\sum_{n=0}^{\infin}}(-1)^n\frac{x^{4n+3}}{(4n+3)(2n+1)!}$

$y=-\frac{1}{S(x)+c}$

$y(-2)=-\frac{1}{-8/3+128/42-2048/1320+c}=1/3$

$1.17+c=3\implies c=1.83$

$y=-\frac{1}{S(x)+1.83}$