
bernoulli equation
xz(dz/dx) = x^2 + 3z^2 with z(1) = 1 ,,,,,,,,,, i have not learned bernoulli's equation and was given this question, although i have read about it on wikipedia, that was not enough can someone give me a step by step explanation of this solution? i do know that i have to perform some sort of substitution but what is that and is there a systematic method to figure out that substitution???? ,,,,,,,,,,,,,,, also my pathetic attempt at this question gave me the answer as ,,,,,,,,,,,,,,, (dy/dx)yx = (x^2)(y^2) + 3 ,,,,,,,,,,,,,,,, i have yet to find the initial value

can't anyone solve this ??????

There's no Bernoulli ODE there, it's just a homogeneous ODE, since $\displaystyle xz\,\frac{dz}{dx}={{x}^{2}}+3{{z}^{2}}\implies \frac{dz}{dx}=\frac{x}{z}+\frac{3z}{x}.$
Put $\displaystyle z=yx,$ or having the initial form of your ODE put $\displaystyle y=z^2.$