Question

Find the general solution: y'=(y/x) + [(x^4)e^-(y/x)^5]/[5y^4]

Answer 1

\[y'= \frac{ y }{ x } + \frac{ x^4 * e ^{-(y/x)^5} }{ 5y^4 }\]

Answer 2

When you say find the general solution, do you mean solve for the original function of x?

Answer 3

Basically it's asking for what y is equal to. Doesn't have to have the form "y=" but just needs the y' to go away.

Answer 4

I can't get it, I wish you luck with this problem!..

Answer 5

This might help a bit. http://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx

Answer 6

One solution is:\[y(x) = x {\log(-5 (const-(\log(x))/5))}^{1/5}\]

Answer 7

\(\color{blue}{\text{Originally Posted by}}\) @SithsAndGiggles
Let \(y=vx\), so \(y'=v+xv'\):
\[\begin{align*}y'=\frac{y}{x}+\frac{x^4}{5y^4\exp{\left(\dfrac{y}{x}\right)^5}}~~\Rightarrow~~v+xv'&=\frac{vx}{x}+\frac{x^4}{5(vx)^4\exp{\left(\dfrac{vx}{x}\right)^5}}\\\\\\
v+xv'&=v+\frac{x^4}{5(vx)^4\exp{v^5}}\\\\\\
xv'&=\frac{1}{5v^4\exp{v^5}}\\\\\\
v^4e^{{\large v^5}}~dv&=\frac{1}{5x}~dx\end{align*}\]
\(\color{blue}{\text{End of Quote}}\)