Question

Solve the separable differential equation for.

dy/dx= [1+x] divided by [xy^15]

Use the following initial condition: y(1)=5

y^16=?

.

Answer 1

Let us solve the separable differential equation

$\frac{dy}{dx}=\frac {1+x}{xy^{15}}$

which is equivalent to

$y^{15}dy=\frac {1+x}{x}dx$

It follows that

$\int y^{15}dy=\int\frac {1+x}{x}dx=\int(\frac{1}x+1)dx.$

Therefore, $\frac{y^{16}}{16}=\ln|x|+x+C.$

Since $y(1)=5,$ we get that $\frac{5^{16}}{16}=1+C,$ ànd hence $C=\frac{5^{16}}{16}-1.$

We conclude that the solution is

$\frac{y^{16}}{16}=\ln|x|+x+\frac{5^{16}}{16}-1$ or

$y^{16}=16\ln|x|+16x+5^{16}-16$