i) In the differential equation [1-(y^2)](dy/dx)+1/y=e^(2x)
make the substitution u=1/y and hence show that the general solution to the equation is y=1/[(Ae^x)-e^(2x)].
Help would be greatly appreciated.
You can bring this differencial equation to an exact differencial equation by an integrating factor.
Write,
(1/y-e^x)+(1-y^2)y'=0
And let,
M(x,y)=1/y-e^x
N(x,y)=1-y^2
Note the cross partial test fails,
M_y not = N_x
But,
(M_y-N_x)/N
Is a function only of "y" thus, you can find an integrating factor.