Learning to Solve a Second Order Differential Equation. We can use the MATLAB command dsolve in order to solve a second order equation.

$($a$)$ Consider the equation

$y^{\text{'}\text{'}}+y^{\text{'}}-6y=20e^t,y(0)=0,y^{\text{'}}(0)=1.$

Solve the equation showing all steps.

$($b$)$ Using MATLAB Proceed as follows,

$>>$ ode$1={?}^{\text{'}}D2y+Dy-6^{**}y={20}^{**}$exp$(t{)}^{\text{'}}$;

$>>$ dsdave$(ode1)$;

then try,

$)=0^{\text{'}},{?}^{\text{'}}Dy(0)=(1^{\text{'}}$

Compare your answers.

$($c$)$ The difficulty we incur with symbolic solvers is that they do not often work for nonlinear equations. Try using dsolve for the following problem and discuss what you notice,

$y^{\text{'}\text{'}}+t^2{y}^{\text{'}}+y=0$

$($d$)$ We therefore find numerical solutions numerically, using the command ode$45.$ In order to do this, we have to create a vector equation, in which we solve for the first$-$derivative and the function itself. We essentially rewrite the equation in the form of $\frac{d}{dt}[y$;$y^{\text{'}}]=[y^{\text{'}},f(t,y,y^{\text{'}})].$ We have MATLAB solve for $y,$ which we will notate in MATLAB as $y(1),$ and $y^{\text{'}}$ which we notate in MATLAB as $y(2).$ For $y^{\text{'}},f(t,y,y^{\text{'}}),$ we define a "anonymous" function. Considering the equation

$y^{\text{'}\text{'}}+t^2{y}^{\text{'}}+y=0,y(0)=1,y^{\text{'}}(0)=-1$

Please use mat lab to answer