This activity is graded for completion so long as a relevant and genuine handwritten attempt is made on each exercise. This activity is due in Gradescope by 11:59PM.

(a)
For each of the following second order linear nonhomogeneous differential equations construct your best guess at the form of a particular solution, using the method of Undetermined Coefficients. You are NOT being asked to solve the equation.
(i)
\(y^{\,\prime \prime }+y=4t^3+3t^2+2t\).
(ii)
\(y^{\,\prime \prime }+y=\sin (t)+\cos (t)\).
(iii)
\(y^{\,\prime \prime }+y=e^{-t}(\sin (t)+\cos (t))\).
(iv)
\(y^{\,\prime \prime }-y^{\,\prime }-2y=e^{2t}\).
(v)
\(y^{\,\prime \prime }-y^{\,\prime }-2y=e^{-2t}\).
(vi)
\(y^{\,\prime \prime }+6y^{\,\prime }+9y=(t-1)e^{3t}\).
(vii)
\(y^{\,\prime \prime }+6y^{\,\prime }+9y=(t-1)e^{-3t}\).
(viii)
\(y^{\,\prime \prime }+2y^{\,\prime }+10y=-4e^{t}\sin {(3t)}\).
(ix)
\(y^{\,\prime \prime }+2y^{\,\prime }+10y=-4e^{-t}\sin {(3t)}\).
(x)
\(y^{\,\prime \prime }+2y^{\,\prime }+10y=\sin {(3t)}\).
(xi)
\(y^{\,\prime \prime \prime }-3y^{\,\prime \prime }+3y^{\,\prime }-y=(1+t^2)e^t\).
(b)
Solve the IVP \(y^{\,\prime \prime }+100y=20\sin {(10t)}\), \(y(0)=0\), \(y^{\,\prime }(0)=0\). Put the answer in the box provided.

(c)
Consider the second order linear nonhomogeneous differential equation \(x^2y^{\,\prime \prime }-3xy^{\,\prime }-5y=f(x)\) on the interval \((0,\infty )\) for the following instances of \(f(x)\) (called a “forcing function".
(i)
Suppose \(f(x)=x^3\). It turns out there is a particular solution of the form \(y_P=Ax^3\). Find it and put the answer in the box provided.

(ii)
Now suppose \(f(x)=\displaystyle \frac {1}{x}\).
i.
Explain why attempting to find a particular solution of the form \(y_P=\dfrac {A}{x}\) is futile. How does it relate to the complementary equation?
ii.
Suppose \(y=uy_1\) is a solution to \(x^2y^{\,\prime \prime }-3xy^{\,\prime }-5y=\displaystyle \frac {1}{x}\), where \(u\) is some function and \(y_1\) is a non-trivial solution to the complementary equation \(x^2y^{\,\prime \prime }-3xy^{\,\prime }-5y=0\). Determine a first order linear nonhomogeneous differential equation that that \(z=u^{\,\prime }\) must satisfy. Put the answer in the box provided.

iii.
Solve that equation for the general solutions for \(z\), then \(u\), and finally \(y\). Put the general solution for \(y\) in the box provided.

(d)
Consider the second order linear nonhomogeneous constant coefficient differential equation \(y^{\,\prime \prime }+y=\sec {(t)}\). The function \(y_1=\cos {(t)}\) is a nontrivial solution to the complementary equation. Suppose \(y=uy_1=u\cos (t)\) is a solution to \(y^{\,\prime \prime }+y=\sec {(t)}\). Determine a differential equation that \(z=u^{\,\prime }\) must satisfy, solve it for the general solution for \(z\), then \(u\), and finally \(y\). Put the general solution for \(y\) in the box provided.