Question

solve (1-x^2) (dy/dx) + xy =x (1-x^2)y^2​

Answer 1

Answer:

Make the substitution vy=1 , so that

0=v′y+vy′=v′y+v⋅xy−y21+x2=v′+v⋅x−y1+x2=v′+x1+x2v−11+x2

This is a linear first-order DE, in the form v′+p(x)v=q(x) , so the following process gives the solution:

Find the integrating factor:

μ(x)=exp(∫p(x)dx)=exp(12ln(1+x2))=1+x2−−−−−√

Multiply by μ(x)dx to make an exact equation; note that μ(x)p(x)=μ′(x) :

μ(x)dv+μ′(x)vdx=μ(x)q(x)dx=dx1+x2√

Integrate both sides:

μ(x)⋅v=∫dx1+x2√=sinh−1(x)+C

Therefore, the solution is y=1+x2−−−−−√sinh−1(x)+C .