Question

Find a function y of x such that

9yy'=x and y(9)=10

Answer 1

The equation $9yy'=x$ is equivalent to $9ydy=xdx.$ It follows that $\int 9ydy=\int xdx,$ and hence the general solution is $\frac{9}2y^2=\frac{1}2x^2+C$ which is equivalent to $9y^2=x^2+C.$

Since $y(9)=10,$ we get that $9\cdot 10^2=9^2+C,$ and hence $C=819.$

We conclude that the particular solution is $9y^2=x^2+819.$