Thread: homogeneous solution??

Hi guys I am really stuck on this differential equation can anyone help?

dy/dt = (t^2 + y^2) / (3ty)

condition - y=1 and t=2

Thank you in advance

This is an example of A DE with Homogeneous Coefficients

ie it can be written in the form dy/dt = f(y/t)
dy/dt = 1/3 t/y + 1/3 y/t

Let z= y/t

tz = y

and z + tdz/dt = dy/dt

z + t dz/dt = 1/(3z) + 1/3z

tdz/dt = 1/(3z) - 2z/3

you now have a separable equation I'll let you go from here

Could I have solved it by turning it into a seperable equation to begin with??

No it's not separable--- the problem being the t^2 + y^2 term.

Typically when fns of y and t are multiplied or divided the eqn can be separated -- when there is addition of fns of y and t you usually cannot separate them.

Go ahead and try but you'll soon see the eqn won't separate

Ok have tried wont work!

From where you finished your explanation do I need to integrate each side? I am not to sure how??

See attachment--you might want to double check my work to see if you agree