Section 1.3 – Classification of Differential Equations
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. Given the following differential equations, classify each as an ordinary differential equation, partial differential equation, give the order. If the equation is an ordinary differential equation, say whether the equation is linear or non linear.
- \(\dfrac{d y}{d x}=3y+x^2\)
- \(5\dfrac{d^4 y}{d x^4}+y=x(x-1)\)
- \(\dfrac{\partial N}{\partial t}=\dfrac{\partial ^2N}{\partial r^2}+\frac{1}{r}\dfrac{\partial N}{\partial r}+kN\)
- \(\dfrac{d x}{d t}=x^2-t\)
- \((1+y^2)y''+ty'+y=e^t\)
\[
\begin{array}{|c|c|c|c|c|}
\hline
Part& ODE& PDE & Order & Linear/Non Linear \\
\hline
(a)&\checkmark & & 1 & Linear\\
\hline
(b)&\checkmark&&4&Linear \\
\hline
(c)&&\checkmark&2&Linear \\
\hline
(d)&\checkmark&&1&Non Linear \\
\hline
(e)&\checkmark&&2&Non Linear\\
\hline
\end{array}
\]
\begin{array}{|c|c|c|c|c|}
\hline
Part& ODE& PDE & Order & Linear/Non Linear \\
\hline
(a)&\checkmark & & 1 & Linear\\
\hline
(b)&\checkmark&&4&Linear \\
\hline
(c)&&\checkmark&2&Linear \\
\hline
(d)&\checkmark&&1&Non Linear \\
\hline
(e)&\checkmark&&2&Non Linear\\
\hline
\end{array}
\]
2. Solve the following.
- Show that \(f(x)=(x^2+Ax+B)e^{-x}\) is a solution to \[y''+2y'+y=2e^{-x}\] for all real numbers \(A\) and \(B\).
- Find a solution that satisfies the initial condition \(y(0)=3\) and \(y'(0)=1\).
- For verification that \(f(x)\) is a solution for part (a), see the video.
- \(y=(x^2+4x+3)e^{-x}\)
3. Determine for which values of \(r\) the function \(t^r\) is a solution of the differential equation
\[t^2y''-4ty'+4y=0, \quad t>0\]
\(r=1,4\)
4. For which values of \(r\) is the function \((x-1)e^{-rx}\) a solution to \(y''-6y'+9y=0?\)
Video Errata: At the 9:46 mark, there is a sign error, we should have \(y = (x − 1)e^{3x}\).
\(r=-3\)