Consider the differential equation $\;\displaystyle x\frac{dy}{dx}+y=\left(xy\right)^2 \ln x\;$ where $y\left(1\right)=\frac{1}{3}$.
Find the exact value of $y\left(e\right)$.
Consider the differential equation $\;\displaystyle \frac{dy}{dx}+y \ln x=x^{-x}e^{-x}\;$ where $y\left(1\right)=0$.
Find the exact value of $y\left(2\right)$.
Consider the differential equation $\;\displaystyle \frac{dy}{dx}=3x-2y+4$ where $y\left(0\right)=1$
Find $y\left(x\right)$.
The particular solution to the differential equation $\displaystyle \frac{dy}{dx}=\frac{4x-y+3}{2x+y-3},\;y\left(0\right)=0\;$ is
$\bigl($
$\bigl)^2 \bigl($
$\bigl)^3=$
*the blanks on the left side of the equation are of the form $ax+by+c$ where $a,b$ and $c$ are integers.
*the blank on the right side of the equation is a number.
The particular solution to the differential equation $\displaystyle \frac{dy}{dx}=\frac{y}{x+\sqrt{xy}}\;$ for $x>0,y>0$ when $y\left(1\right)=2$ is