Thread: First order ODE involving substitution

Hey guys, I'm new here, rather stuck on this IVP. Full working would be lovely as I haven't come across substitution before and have attempted this problem about 7 times so far..

By making the substitution z = 3x - y, show that solving the initial value problem

y' = y -3x -5 -1/(3x -y +8), y(0) = 0,

leads to

(3x -y +8)^2 = 65e^2x -1.

Any help guys? I've tried this problem multiple times, probably doesn't help that I haven't been taught how substitutions come into ODE's.. Many thanks in advance

Using the substitution suggested will make the equation more manageable.



using gives

I see that, but my problem is I can't solve the equation from there, each time I try to solve it I get a different large mess of working that goes nowhere.

Originally Posted by pickslides

Using the substitution suggested will make the equation more manageable.



using gives

The substitution must be complete and that means that so that the ODE in z is...

(1)

... where the variables z and x are separated...

Kind regards