Problem 15
\(\left(2 e^{y}-x\right) y^{\prime}=1, y(0)=0,[-5,5]\). Hint: dsolve \(\left({ }^{\prime}(2 * \exp (\mathrm{y})-\mathrm{x}) * \mathrm{Dy}=1^{\prime}, \mathrm{y}^{\prime}(0)=0^{\prime}, \mathrm{x}^{\prime}\right)\).
Problem 16
\(\left(x+y^{2}\right) y^{\prime}=y, y(0)=4,[-4,6]\).
Problem 18
In Exercises 18 and 19, use dsolve to obtain the solution of each of the indicated second order differential equations. Use the simple command to find the simplest form of that solution. Use ezplot to sketch the solution on the indicated time interval. $$ \begin{aligned} &y^{\prime \prime}+4 y=3 \cos 2.1 t \\ &y(0)=0, y^{\prime}(0)=0,[0,64 \pi] \end{aligned} $$
Problem 19
In Exercises 18 and 19, use dsolve to obtain the solution of each of the indicated second order differential equations. Use the simple command to find the simplest form of that solution. Use ezplot to sketch the solution on the indicated time interval. $$ \begin{aligned} &y^{\prime \prime}+16 y=3 \sin 4 t \\ &y(0)=0, y^{\prime}(0)=0,[0,32 \pi] \end{aligned} $$
Problem 20
Use dsolve to obtain the solution of \(y^{\prime \prime}+y^{\prime}+144 y=\cos 2 t, y(0)=0, y^{\prime}(0)=0\). Use ezplot to plot the solution on the time interval \([0,6 \pi]\). Use ode45 to find a numerical solution on the same time interval and superimpose that plot of the first plot. How closely does the numerical solution match the symbolic solution?
Problem 22
Use dsolve to solve the initial value problem $$ \begin{aligned} &y^{\prime}=v \\ &v^{\prime}=-2 v-2 y+\cos 2 t \end{aligned} $$ with \(y(0)=1\) and \(v(0)=-1\), over the time interval \([0,30]\).
Problem 23
In Exercises \(23-25\), find the solutions to the indicated initial value problem. \(t^{2} y^{\prime \prime}=\left(y^{\prime}\right)^{2}\), with \(y(1)=3\) and \(y^{\prime}(1)=2\).
Problem 24
\(y^{\prime \prime}=y y^{\prime}\), with \(y(0)=0\), and \(y^{\prime}(0)=2\).
Problem 25
\(y^{\prime \prime}=t y^{\prime}+y+1\), with \(y(0)=1\), and \(y^{\prime}(0)=0\).
