Euler-homogeneous problem?

Question

Can someone explain me how to do this problem with Euler-homogeneous and Apply this method to solve the initial-value problem

dy/dt=(y^2+t^2)/(yt) y(1)=-2 t>0

I've tried to change the equation into this

dy/dt=y/t + t/y

but im not sure if thats right? please if you know how to do this I will thank you a lot.

Answer 1

Yes, that's right. Now note that if u=y/t, du/dt = (dy/dt t - y)/t² = (dy/dt - u)/t, so dy/dt = t du/dt + u. Now making the substitution:

t du/dt + u = u+1/u

Subtracting u from both sides:

t du/dt = 1/u

Multiplying by u/t:

u du/dt = 1/t

Now integrating both sides with respect to t:

u²/2 = ln t + C
u² = 2 ln t + C

Making the reverse substitution:

y²/t² = 2 ln t + C
y² = 2t² ln t + Ct²
y = ±t√(2 ln t + C)

Now, we must find C. Using y(1)=-2:

-2 = ±√C

So we see that the negative sign of the function must be taken, and squaring both sides, we see that:

C = 4

So y = -t√(2 ln t + 4)