Question

Solve x-yp=ap^2...solve by differential equation

Answer 1

Answer:

Solve x+yp=ap2 where p=dydx

My Attmept:

x+yp=ap2

x=ap2−yp

This is solvable for x so differentiating both sides w.r.t y

dxdy=2ap⋅dpdy−y⋅dpdy−p

1p+p=(2ap−y)⋅dpdy

1+p2p=(2ap−y)⋅dpdy

dydp=2ap21+p2−yp1+p2

dydp+p1+p2⋅y=2ap21+p2

This is Linear so

Integrating Factor=e∫p1+p2dp=1+p2−−−−−√

So we have

y1+p2−−−−−√=∫2ap21+p2×1+p2−−−−−√dp+c

y1+p2−−−−−√=2a∫p21+p2−−−−−√dp+c

I could not solve further from here

y1+p2−−−−−√=ap1+p2−−−−−√−asinh−1p+C⟹y(p)=ap+C−sinh−1p1+p2−−−−−√.

put this in

x(p)=ap2−py(p)

to get x(p).

Finally x(p) and y(p) constitue the parametric solution of the ODE, where p acts as a parameter only.C is the integration-constant,

Note that

∫p21+p2−−−−−√dp=∫(1+p2−−−−−√−11+p2−−−−−√)dp=12(p1+p2−−−−−√−sinh−1p)

Both the integrals are standard ones.

Oh the sign before sinh−1p should have been -, i have corrected it now and given how to evaluate the required integral.

Usually you get fastest to the relevant equations if you start with the derivative by p, using y˙(p)=px˙(p). Then

x˙+py˙+y=2ap

Multiply with p and eliminate x

(1+p2)y˙+py=2ap2⟹1+p2−−−−−√y(p)=∫2ap21+p2−−−−−√dp

Now substitute p=sinh(u) in the last integral,

=∫a(cosh(2u)−1)du=asinh(u)cosh(u)−au+C=ap1+p2−−−−−√−asinh−1(p)