Question

Solve differential equation \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{x}}}{{{y}}}},\ {y}{\left({0}\right)}=-{8}\)

Answer 1

\(\displaystyle{y}{\left.{d}{y}\right.}={x}{\left.{d}{x}\right.}\)
\(\displaystyle\int{y}{\left.{d}{y}\right.}=\int{x}{\left.{d}{x}\right.}\)
\(\displaystyle{\frac{{{y}^{{2}}}}{{{2}}}}={\frac{{{x}^{{2}}}}{{{2}}}}+{c}\)
Now we use initial condition
\(\displaystyle{y}{\left({0}\right)}=-{8}\)
Substitute the value we get
\(\displaystyle{\frac{{{\left(-{8}\right)}^{{2}}}}{{{2}}}}={\frac{{{0}^{{2}}}}{{{2}}}}+{c}\)
\(\displaystyle{\frac{{{64}}}{{{2}}}}={c}\)
\(\displaystyle{c}={32}\)
\(\displaystyle{\frac{{{y}^{{2}}}}{{{2}}}}={\frac{{{x}^{{2}}}}{{{2}}}}+{32}\)
\(\displaystyle{y}^{{2}}={x}^{{2}}+{64}\)