Exercises

(1)
Which is NOT a solution to the second order linear homogeneous constant-coefficient differential equation \(\displaystyle y\,^{\prime \prime }-y\,^{\prime }-12y=0\)?
(a)
\(y_1=\dfrac {1}{e^{-1-4t}}\)
(b)
\(y_2=\dfrac {2}{e^{3t}}\)
(c)
\(y_3=e^{1-3t}-2e^{1+4t}\)
(d)
\(y_4=e^{1-3t}-3e^{1-4t}\)
(e)
\(y_5=-4e^{1+4t}\)
(f)
\(y_6=0\)
(2)
Suppose \(y(t)\) satisfies \(y^{\,\prime \prime }+2y^{\,\prime }+10y=0\), \(y(0)=-1\) and \(y^{\,\prime }(0)=1\). Determine \(y(\pi )\). Round to the nearest hundredth.
(3)
Suppose \(y(t)\) satisfies \(9y^{\,\prime \prime }+6y^{\,\prime }+y=0\), \(y(0)=3\), \(y^{\,\prime }(0)=0\). Find \(\displaystyle \lim _{t\rightarrow \infty } y(t)\). Round to the nearest hundredth.
(4)
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 compute the Wronskian: the Wronskian of \(\{f,g,h\}\) is
\(\displaystyle W(x)=\left |\begin{array}{ccc} f & g & h\\ f' & g' & h'\\ f'' & g'' & h''\end{array}\right | = f\left |\begin{array}{cc} g' & h'\\ g'' & h'' \end{array}\right |-g\left |\begin{array}{cc} f' & h'\\ f'' & h'' \end{array}\right | + h\left |\begin{array}{cc} f' & g'\\ f'' & g'' \end{array}\right |\).
Suppose the functions \(y_1,y_2,y_3\) are solutions to \(y\,^{\prime \prime \prime }+py\,^{\prime \prime }+qy\,^{\prime }+ry=0\) on an open interval \(I\) on which \(p,q,r\) are continuous functions. \(\{y_1,y_2,y_3\}\) is a fundamental set of solutions (meaning the general solution is
\(c_1 y_1+c_2 y_2+c_3 y_3\)) if and only if their Wronskian is not \(0\). Find all solutions of the form \(e^{rt}\) and \(te^{rt}\) for \(y\,^{\prime \prime \prime }+y\,^{\prime \prime }-y\,^{\prime }-y=0\), and compute their Wronskian, justifying that they form a fundamental set of solutions to this third-order linear homogeneous differential equation. Determine \(|W(2)|\). Round to the nearest hundredth.
(5)
Suppose \(y(t)\) satisfies \(y\,^{\prime \prime \prime }+y\,^{\prime \prime }-2y=0\), \(y(0)=0\), \(y'(0)=2\), \(y^{\,\prime \prime }(0)=1\). Determine \(y(3)\). Round to the nearest thousandth.

Hint: \(r^3+r^2-2=(r-1)(r^2+2r+2)\).