2.6: Problems

Last updated
Save as PDF

Page ID: 91056

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Find all of the solutions of the second order differential equations. When an initial condition is given, find the particular solution satisfying that condition.
1. \(y^{\prime \prime}-9 y^{\prime}+20 y=0\)
2. \(y^{\prime \prime}-3 y^{\prime}+4 y=0, \quad y(0)=0, \quad y^{\prime}(0)=1\).
3. \(8 y^{\prime \prime}+4 y^{\prime}+y=0, \quad y(0)=1, \quad y^{\prime}(0)=0\).
4. \(x^{\prime \prime}-x^{\prime}-6 x=0\) for \(x=x(t)\).
Verify that the given function is a solution and use Reduction of Order to find a second linearly independent solution.
1. \(x^{2} y^{\prime \prime}-2 x y^{\prime}-4 y=0, \quad y_{1}(x)=x^{4}\).
2. \(x y^{\prime \prime}-y^{\prime}+4 x^{3} y=0, \quad y_{1}(x)=\sin \left(x^{2}\right)\).
3. \(\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+2 y=0, \quad y_{1}(x)=x\). [Note: This is one solution of Legendre’s differential equation in Example 4.4.]
4. \((x-1) y^{\prime \prime}-x y^{\prime}+y=0, \quad y_{1}(x)=e^{x} .\)
Prove that \(y_{1}(x)=\sinh x\) and \(y_{2}(x)=3 \sinh x-2 \cosh x\) are linearly independent solutions of \(y^{\prime \prime}-y=0 .\) Write \(y_{3}(x)=\cosh x\) as a linear combination of \(y_{1}\) and \(y_{2}\).
Consider the nonhomogeneous differential equation \(x^{\prime \prime}-3 x^{\prime}+2 x=6 e^{3 t}\).
1. Find the general solution of the homogenous equation.
2. Find a particular solution using the Method of Undetermined Coefficients by guessing \(x_{p}(t)=A e^{3 t}\).
3. Use your answers in the previous parts to write down the general solution for this problem.
Find the general solution of the given equation by the method given.
1. \(y^{\prime \prime}-3 y^{\prime}+2 y=10\), Undetermined Coefficients.
2. \(y^{\prime \prime}+2 y^{\prime}+y=5+10 \sin 2 x\), Undetermined Coefficients.
3. \(y^{\prime \prime}-5 y^{\prime}+6 y=3 e^{x}\), Reduction of Order.
4. \(y^{\prime \prime}+5 y^{\prime}-6 y=3 e^{x}\), Reduction of Order.
5. \(y^{\prime \prime}+y=\sec ^{3} x\), Reduction of Order.
6. \(y^{\prime \prime}+y^{\prime}=3 x^{2}\), Variation of Parameters.
7. \(y^{\prime \prime}-y=e^{x}+1\), Variation of Parameters.
Use the Method of Variation of Parameters to determine the general solution for the following problems.
1. \(y^{\prime \prime}+y=\tan x\).
2. \(y^{\prime \prime}-4 y^{\prime}+4 y=6 x e^{2 x}\).
3. \(y^{\prime \prime}-2 y^{\prime}+y=\dfrac{e^{2 x}}{\left(1+e^{x}\right)^{2}}\).
4. \(y^{\prime \prime}-3 y^{\prime}+2 y=\cos \left(e^{x}\right)\).
Instead of assuming that \(c_{1}^{\prime} y_{1}+c_{2}^{\prime} y_{2}=0\) in the derivation of the solution using Variation of Parameters, assume that \(c_{1}^{\prime} y_{1}+c_{2}^{\prime} y_{2}=h(x)\) for an arbitrary function \(h(x)\) and show that one gets the same particular solution.
Find all of the solutions of the second order differential equations for \(x>\) 0. When an initial condition is given, find the particular solution satisfying that condition.
1. \(x^{2} y^{\prime \prime}+3 x y^{\prime}+2 y=0\).
2. \(x^{2} y^{\prime \prime}-3 x y^{\prime}+3 y=0, \quad y(1)=1, y^{\prime}(1)=0\).
3. \(x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0\).
4. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+3 y=0, \quad y(1)=3, y^{\prime}(1)=0\).
5. \(x^{2} y^{\prime \prime}+3 x y^{\prime}-3 y=0\).
Another approach to solving Cauchy-Euler equations is by transforming the equation to one with constant coefficients.
1. \[\dfrac{d^{2} y}{d x^{2}}=\dfrac{1}{x^{2}}\left(\dfrac{d^{2} v}{d t^{2}}-\dfrac{d v}{d t}\right) \nonumber \]
2. Use the above transformation to solve the following equations:
  1. \(x^{2} y^{\prime \prime}+3 x y^{\prime}-3 y=0\).
  2. \(2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=0\).
  3. \(4 x^{2} y^{\prime \prime}+y=0\).
  4. \(x^{3} y^{\prime \prime \prime}+x y^{\prime}-y=0\).
Solve the following nonhomogenous Cauchy-Euler equations for \(x>0\).
1. \(x^{2} y^{\prime \prime}+3 x y^{\prime}-3 y=3 x^{2}\).
2. \(2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=x^{2}+x\).
3. \(x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=2 x^{3}\).
4. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+3 y=5 x^{2}, \quad y(1)=3, y^{\prime}(1)=0\).
A spring fixed at its upper end is stretched six inches by a 1o-pound weight attached at its lower end. The spring-mass system is suspended in a viscous medium so that the system is subjected to a damping force of \(5 \dfrac{d x}{d t}\) lbs. Describe the motion of the system if the weight is drawn down an additional 4 inches and released. What would happen if you changed the coefficient " 5 " to " 4 "? [You may need to consult your introductory physics text. For example, the weight and mass are related by \(W=m g\), where the mass is in slugs and \(g=32 \mathrm{ft} / \mathrm{s}^{2}\).]
Consider an LRC circuit with \(L=1.00 \mathrm{H}, R=1.00 \times 10^{2} \Omega, C=\) \(1.00 \times 10^{-4} \mathrm{f}\), and \(V=1.00 \times 10^{3}\) V. Suppose that no charge is present and no current is flowing at time \(t=0\) when a battery of voltage \(V\) is inserted. Find the current and the charge on the capacitor as functions of time. Describe how the system behaves over time.
Consider the problem of forced oscillations as described in section \(2.4.2.\)
1. Plot the solutions in Equation \((2.4.35)\) for the following cases: Let \(c_{1}=0.5, c_{2}=0, F_{0}=1.0 \mathrm{~N}\), and \(m=1.0 \mathrm{~kg}\) for \(t \in[0,100] .\)
  1. \(\omega_{0}=2.0 \mathrm{rad} / \mathrm{s}, \omega=0.1 \mathrm{rad} / \mathrm{s}\).
  2. \(\omega_{0}=2.0 \mathrm{rad} / \mathrm{s}, \omega=0.5 \mathrm{rad} / \mathrm{s}\).
  3. \(\omega_{0}=2.0 \mathrm{rad} / \mathrm{s}, \omega=1.5 \mathrm{rad} / \mathrm{s}\).
  4. \(\omega_{0}=2.0 \mathrm{rad} / \mathrm{s}, \omega=2.2 \mathrm{rad} / \mathrm{s}\).
  5. \(\omega_{0}=1.0 \mathrm{rad} / \mathrm{s}, \omega=1.2 \mathrm{rad} / \mathrm{s}\).
  6. \(\omega_{0}=1.5 \mathrm{rad} / \mathrm{s}, \omega=1.5 \mathrm{rad} / \mathrm{s}\).
2. Confirm that the solution in Equation \((2.4.36)\) is the same as the solution in Equation \((2.4.35)\) for \(F_{0}=2.0 \mathrm{~N}, m=10.0 \mathrm{~kg}, \omega_{0}=1.5\) \(\mathrm{rad} / \mathrm{s}\), and \(\omega=1.25 \mathrm{rad} / \mathrm{s}\), by plotting both solutions for \(t \in\) \([0,100]\).
A certain model of the motion light plastic ball tossed into the air is given by

\(m x^{\prime \prime}+c x^{\prime}+m g=0, \quad x(0)=0, \quad x^{\prime}(0)=v_{0}\)

Here \(m\) is the mass of the ball, \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) is the acceleration due to gravity and \(c\) is a measure of the damping. Since there is no \(x\) term, we can write this as a first order equation for the velocity \(v(t)=x^{\prime}(t)\) :

\(m v^{\prime}+c v+m g=0\)

Find the general solution for the velocity \(v(t)\) of the linear first order differential equation above.
Use the solution of part a to find the general solution for the position \(x(t)\).
Find an expression to determine how long it takes for the ball to reach it’s maximum height?
Assume that \(c / m=5 \mathrm{~s}^{-1}\). For \(v_{0}=5,10,15,20 \mathrm{~m} / \mathrm{s}\), plot the solution, \(x(t)\) versus the time, using computer software.
From your plots and the expression in part \(c\), determine the rise time. Do these answers agree?
What can you say about the time it takes for the ball to fall as compared to the rise time?

Find the solution of each initial value problem using the appropriate initial value Green’s function.
1. \(y^{\prime \prime}-3 y^{\prime}+2 y=20 e^{-2 x}, \quad y(0)=0, \quad y^{\prime}(0)=6\)
2. \(y^{\prime \prime}+y=2 \sin 3 x, \quad y(0)=5, \quad y^{\prime}(0)=0\).
3. \(y^{\prime \prime}+y=1+2 \cos x, \quad y(0)=2, \quad y^{\prime}(0)=0\).
4. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=3 x^{2}-x, \quad y(1)=\pi, \quad y^{\prime}(1)=0\).
Use the initial value Green’s function for \(x^{\prime \prime}+x=f(t), x(0)=4\), \(x^{\prime}(0)=0\), to solve the following problems.
1. \(x^{\prime \prime}+x=5 t^{2}\).
2. \(x^{\prime \prime}+x=2 \tan t\).
For the problem \(y^{\prime \prime}-k^{2} y=f(x), y(0)=0, y^{\prime}(0)=1\),
1. Find the initial value Green’s function.
2. Use the Green’s function to solve \(y^{\prime \prime}-y=e^{-x}\).
3. Use the Green’s function to solve \(y^{\prime \prime}-4 y=e^{2 x}\).
Find and use the initial value Green’s function to solve

\(x^{2} y^{\prime \prime}+3 x y^{\prime}-15 y=x^{4} e^{x}, \quad y(1)=1, y^{\prime}(1)=0\)