Problem Set #$1-$ Exact ODEs

For each ODE in the table:

i$.)$ Prove whether the ODE is exact. If the ODE is not exact, use an integrating factor to make it exact $($see p$.25$ in book for help$).$

ii$.)$ Derive the general solution to the exact ODE.

iii.$)$ Find the particular solution given the known conditions.

$\backslash$table$[[$Differential equation,Known condition$]]$

$1$a$.)2xln(y)dx\frac{x^2}{y}dy=0,y(1)=1$

$1$b$.),x^{3}dx{y}^{3}dy=0,y(0)=2$

$1$c$.),sinxcosydxcosxsinydy=0,y(0)=3$

$1$d$.),dy-y^{2}sinxdx=0,y(1)=4$

$1$e$.)(3x{e}^{y}2y)dx(x^2{e}^{y}x)dy=0,y(1)=5$