Exercise$1$:

Solve the following homogeneous ODE on $(0,)$

i$)\frac{dy}{dx}=\frac{y^3+x{y}^2}{y{x}^2-x^3}.$

ii$)\frac{dy}{dx}=\frac{5x-3y}{3x+5y}.$

iii$)2(x-y)e^{\frac{y}{x}}dx+x(1+2e^{\frac{y}{x}})dy=0.$

Exercise$2$ :

Solve the ODE

i$)\frac{dy}{dx}+\frac{xy}{1+x^2}=x\sqrt[\phantom{2}]{y}.$

ii$)\frac{dy}{dx}+\frac{y}{x}=\frac{y^2}{x^2}.$

Exercise $3$ :

Verify that the following differential equation is exact and solve it

$(sin(xy)+$xycos$(xy))dx+x^{2}cos(xy)dy=0$

Find the values of parameters a and $b$ for which the ODE

$(a{x}^{2}y+y^3)dx+(\frac{1}{3}{x}^3+bx{y}^2)dy=0$

is exact, and then solve it in the case of those values of a and $b.$

Exercise$4$:

Solve the following

D$.$Es by finding an appropriate integrating factor

i$)ydx+(2xy-e^{-2y})dy=0.$

ii$)(3y^3-1)dx+9y^2(1-3e^{-x})dy=0.$

Exercise$5$:

Find the Orthogonal trajectories of the family of curves

i$)x^2=4c{y}^3$

ii$)x^2+3y^2=cy,$

where c is a constant.