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 homogeneous differential equations, show that the given functions \(y_1,y_2\) are solutions. Then compute the Wronskian \(W=y_1y_2'-y_1'y_2\) to determine if they form a fundamental set of solutions or not. If they do form a fundamental set of solutions, write down the general solution as \(y=c_1y_1+c_2y_2\).
(i)
\(y^{\,\prime \prime }-3y^{\,\prime }+2y=0\); \(y_1=e^t\); \(y_2=e^{2t}\)
(ii)
\(y^{\,\prime \prime }-8y^{\,\prime }+16y=0\); \(y_1=e^{4t}\); \(y_2=te^{4t}\)
(iii)
\(y^{\,\prime \prime }-y=0\), \(y_1=e^{1-t}\); \(y_2=2e^{-t}\)
(iv)
\(y^{\,\prime \prime }+4y=0\); \(y_1=\cos {(2t)}\); \(y_2=\sin {(2t)}\)
(v)
\(y^{\,\prime \prime }+2y^{\,\prime }+5y=0\); \(y_1=e^{-t}\cos {(2t)}\); \(y_2=e^{-t}\sin {(2t)}\)
(vi)
\(y^{\,\prime \prime }+\dfrac {1}{t}y^{\,\prime }-\dfrac {1}{t^2}y=0\); \(y_1=t\); \(y_2=\dfrac {1}{t}\)

For a third-order linear homogeneous differential equation, we need three linearly independent solutions to form a fundamental set of solutions. In order to check the linear independence of a given set of solutions, we can compute the Wronskian: The Wronskian of \(f,g,h\) is

\[ W(x)=\left |\begin{array}{ccc} f & g & h\\ f' & g' & h'\\ f'' & g'' & h''\end{array}\right | = f\left |\begin{array}{cc} g' & h'\\ g'' & h'' \end{array}\right |-g\left |\begin{array}{cc} f' & h'\\ f'' & h'' \end{array}\right | + h\left |\begin{array}{cc} f' & g'\\ f'' & g'' \end{array}\right | \]

where the absolute value bars denote determinants.

(b)
Consider \(y^{\,\prime \prime \prime }+3y^{\,\prime \prime }+3y^{\,\prime }+y=0\)
(i)
Check that \(y_{1}=e^{-t}\), \(y_2=te^{-t}\) and \(y_3=t^2e^{-t}\) are solutions.
(ii)
Compute the Wronskian of the solutions you found to determine if they form a fundamental set of solutions or not. If they do form a fundamental set of solutions, write down the general solution as \(y=c_1y_1+c_2y_2+c_3y_3\).
(c)
Consider \(y^{\,\prime \prime \prime }-y^{\,\prime \prime }+9y^{\,\prime }-9y=0\)
(i)
Check that \(y_{1}=e^{t}\), \(y_2=\cos {(3t)}\) and \(y_3=\sin {(3t)}\) are solutions.
(ii)
Compute the Wronskian of the solutions you found to determine if they form a fundamental set of solutions or not. If they do form a fundamental set of solutions, write down the general solution as \(y=c_1y_1+c_2y_2+c_3y_3\).
(d)
Consider \(y^{(4)}+2y^{\,\prime \prime }+y=0\). Show that \(y_1=\cos {(t)}\), \(y_2=\sin {(t)}\), \(y_3=t\cos {(t)}\), and \(y_4=t\sin {(t)}\) are each solutions. Given that these form a fundamental set of solutions, write down the general solution.
(e)
Given three linearly independent solutions \(y_1,y_2,y_3\) to a third order linear homogeneous differential equation, for any solution \(y\) to that same equation, the functions \(y_1,y_2,y_3,y\) must form a linearly dependent set, which implies (via linear algebra) that

\(\displaystyle \mathrm {det}\left (\begin{array}{cccc} y_1 & y_2 & y_3 & y\\ y_1' & y_2' & y_3' & y'\\ y_1'' & y_2'' & y_3'' & y'' \\ y_1''' & y_2''' & y_3''' & y''' \end{array}\right ) \)

\[= y_1\,\mathrm {det}\left (\begin{array}{ccc} y_2' & y_3' & y'\\ y_2'' & y_3'' & y''\\ y_2''' & y_3''' & y'''\end{array}\right )-y_2\,\mathrm {det}\left (\begin{array}{ccc} y_1' & y_3' & y'\\ y_1'' & y_3'' & y''\\ y_1''' & y_3''' & y''' \end{array}\right ) + y_3\,\mathrm {det}\left (\begin{array}{ccc} y_1' & y_2' & y'\\ y_1'' & y_2'' & y''\\ y_1''' & y_2''' & y'''\end{array}\right )-y\,\mathrm {det}\left (\begin{array}{ccc} y_1' & y_2' & y_3'\\ y_1'' & y_2'' & y_3''\\ y_1''' & y_2''' & y_3'''\end{array}\right )=0.\]

Use this to construct a third order linear homogeneous differential equation for which \(y_1=x\), \(y_2=x^2\) and \(y_3=e^x\) are linearly independent solutions.