Problem #$3$: Solve the following initial value problem analytically $($by hand$).$

$y^{\text{'}}=\frac{1}{3}y-e^{-7x},y(0)=7$

Enter your answer as a

$\frac{3}{{22}^{**}}(e^{-7^{**}x})+(\frac{151}{{22}^{**}}{e}^{\frac{x}{3}})$

$\frac{3}{22}{e}^{-7x}+\frac{151}{22}{e}^{\frac{x}{3}}$

symbolic function $ofx,asin$

these examples

Problem #$3$:

Do NOT include $\text{'}y=\text{'}$ in your

answer.Problem # $4$: Produce a direction field for the differential equation in Problem #$3$ above for values

of $x$ in the interval $-2,2$ and values of $y$ in the interval $5,9$ using the default blue

color for the arrows. The values for $x$ and $y$ should be $0\mathrm{.}2$ apart. Normalize the

arrows in the direction field so that they are of unit length, and use the same scale on

both coordinate axes.

Then plot $($on the same graph$)$ the exact solution curve from #$3$ above. Use red for

the solution curve, and a LineWidth of $2($type help plot for information about

the LineWidth command.$)$

Your axes for the combined plot should be the same as those specified for the

direction field $($i$.$e$.,-2,2$ and $5,9).$ To achieve this you might need to use the

ylim or xlim commands. $($Type help ylim or help xlim for details.$)$ But

note that you will not be able to use the $x$ from the meshgrid function for your

exact curve, because that $x$ is a matrix. So you will have to produce a new $x$ vector

for use in the exact plot.

Use your First Name, Last Name, and Student Number as the title for the graph $($e$.$g$.,$

'Johnny Good, $1234567\text{'}).$ Then save your graph as a Portable Network Graphics

$(.$png$)$ file, and upload it$.$

PLEASE USE MATLAB FOR THIS