Activity $2$

Forrogeneous ODE

Solve the following differential equations.

a$)y^{\text{'}}=\frac{4x^2+y^2}{xy}$

b$)x{y}^{\text{'}}=\frac{1}{2}{y}^2+y$

c$)2xy{y}^{\text{'}}=y^2-x^2$

d$)2xy{y}^{\text{'}}=3y^2+x^2,$ the initial value, $y(1)=2$

e$)y^{\text{'}}=\frac{y}{x}+(\frac{2x^3}{y})cos(x^2)$ the initial value, $y(\sqrt[\phantom{2}]{\frac{}{2}})=\sqrt[\phantom{2}]{}$

f$)x{y}^{\text{'}}=y+4x^{5}co{s}^2(\frac{y}{x})$ the initial value, $y(2)=0$

Answer:

$($a$)y^2=x^{2}8ln|x|+C$

$($b$)y=\frac{2x}{C-x}$

$($c$)x^2+y^2=x{e}^C$

$($d$)y=+-x\sqrt[\phantom{2}]{5x-1}$

$($e$)y=x\sqrt[\phantom{2}]{2sin(x^2)}$

$($f$)y=xta{n}^{-1}(x^4-16)$