Answer
Picture Attached
Work Step by Step
The question asks for a plot of the slope field with a computer algebra system, which is pictured above. It also asks to plot the solution to the following differential equation that satisfies $y(0) = 2$:
$\frac{dy}{dx} = \frac{x}{y}*e^{\frac{x}{8}}$
This differential equation can be solved using separation of variables in the following way:
$\frac{dy}{dx} = \frac{x}{y}*e^{\frac{x}{8}} \implies dy = \frac{x}{y}*e^{\frac{x}{8}} dx \implies y$ $dy = x e^{\frac{x}{8}} $ $dx $
Now, we integrate both sides:
$\int y$ $dy = \int x e^{\frac{x}{8}} $ $dx $ The left side can be integrated using the power rule, for the right side we will use integration by parts.
$\implies \frac{y^{2}}{2} = \int x e^{\frac{x}{8}} $ $dx $
Letting $u = x , v' = e^{\frac{x}{8}}$, we integrate by parts:
$\implies \int x e^{\frac{x}{8}} $ $dx =8xe^{\frac{x}{8}} - \int 8e^{\frac{x}{8}}$ $dx $ $= 8xe^{\frac{x}{8}} - 64e^{\frac{x}{8}} + C = 8e^{\frac{x}{8}}(x-8) + C$
$\implies \frac{y^{2}}{2} = 8e^{\frac{x}{8}}(x-8) + C \implies y = \sqrt{16e^{\frac{x}{8}}(x-8) + 2C}$
Now, we impose the boundary condition of $y(0) = 2 \implies 2 = \sqrt{16e^{\frac{0}{8}}(0-8) + 2C} = \sqrt{-124 + 2C} $
$\implies 4 = -124 + 2C \implies C =66$
Thus, the plotted solution is:
$y = \sqrt{16e^{\frac{x}{8}}(x-8) + 132}$
