If (dx)/(dy)=1+x-y^2y,x(1)=1, then 5x(2) is equal to __________.

Question

If $\frac{dx}{dy}=\frac{1+x-{y}^{2}}{y},x(1)=1$, then $5x(2)$ is equal to \_\_\_\_\_\_\_\_\_\_.

Answer: 5

Accepted Answer

$\frac{dx}{dy}-\frac{x}{y}=\frac{1-{y}^{2}}{y}$

Integrating factor $={e}^{\int -\frac{1}{y}dy}=\frac{1}{y}$

$\begin{gathered} x\cdot \frac{1}{y}=\int \frac{1-{y}^{2}}{{y}^{2}}dy \ \frac{x}{y}=\frac{-1}{y}-y+c \ x=-1-{y}^{2}+cy \end{gathered}$

$\begin{gathered} x(1)=1 \ 1=-1-1+c\Rightarrow c=3 \ x=-1-{y}^{2}+3y \ 5x(2)=5(-1-4+6) \ =5 \end{gathered}$

Solutions · 1