2.3E: Exercises
- Page ID
- 17367
This page is a draft and is under active development.
In Exercises \((2.3E.1)\) to \((2.3E.6)\), find a particular solution by the method used in Example \((2.3.2)\). Then find the general solution and, where indicated, solve the initial value problem and graph the solution.
Exercise \(\PageIndex{1}\)
\(y''+5y'-6y=22+18x-18x^2\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{2}\)
\(y''-4y'+5y=1+5x\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{3}\)
\(y''+8y'+7y=-8-x+24x^2+7x^3\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{4}\)
\(y''-4y'+4y=2+8x-4x^2\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{5}\)
\(y''+2y'+10y=4+26x+6x^2+10x^3, \quad y(0)=2, \quad y'(0)=9\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{6}\)
\(y''+6y'+10y=22+20x, \quad y(0)=2,\; y'(0)=-2\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{7}\)
Show that the method used in Example \((2.3.2)\) won't yield a particular solution of
\begin{equation}\label{eq:2.3E.1}
y''+y'=1+2x+x^2;
\end{equation}
that is, \eqref{eq:2.3E.1} doesn't have a particular solution of the form \(y_p=A+Bx+Cx^2\), where \(A\), \(B\), and \(C\) are constants.
- Answer
-
Add texts here. Do not delete this text first.
In Exercises \((2.3E.8)\) to \((2.3E.13)\), find a particular solution by the method used in Example \((2.3.3)\).
Exercise \(\PageIndex{8}\)
\(x^2y''+7xy'+8y=\displaystyle{6\over x}\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{9}\)
\(x^2y''-7xy'+7y=13x^{1/2}\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{10}\)
\(x^2y''-xy'+y=2x^3\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{11}\)
\(x^2y''+5xy'+4y=\displaystyle{1\over x^3}\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{12}\)
\(x^2y''+xy'+y=10x^{1/3}\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{13}\)
\(x^2y''-3xy'+13y=2x^4\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{14}\)
Show that the method suggested for finding a particular solution in Exercises \((2.3E.8)\) to \((2.3E.13)\) won't yield a particular solution of
\begin{equation}\label{eq:2.3E.2}
x^2y''+3xy'-3y={1\over x^3};
\end{equation}
that is, \eqref{eq:2.3E.2} doesn't have a particular solution of the form \(y_p=A/x^3\).
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{15}\)
Prove: If \(a\), \(b\), \(c\), \(\alpha\), and \(M\) are constants and \(M\ne0\) then
\begin{eqnarray*}
ax^2y''+bxy'+cy=M x^\alpha
\end{eqnarray*}
has a particular solution \(y_p=Ax^\alpha\) (\(A=\) constant) if and only if \(a\alpha(\alpha-1)+b\alpha+c\ne0\).
- Answer
-
Add texts here. Do not delete this text first.
If \(a\), \(b\), \(c\), and \(\alpha\) are constants, then
\begin{eqnarray*}
a(e^{\alpha x})''+b(e^{\alpha x})'+ce^{\alpha x}=(a\alpha^2+b\alpha+c)e^{\alpha x}.
\end{eqnarray*}
Use this in Exercises \((2.3E.16)\) to \((2.3E.21)\) to find a particular solution. Then find the general solution and, where indicated, solve the initial value problem and graph the solution.
Exercise \(\PageIndex{16}\)
\(y''+5y'-6y=6e^{3x}\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{17}\)
\(y''-4y'+5y=e^{2x}\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{18}\)
\(y''+8y'+7y=10e^{-2x}, \quad y(0)=-2,\; y'(0)=10\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{19}\)
\(y''-4y'+4y=e^{x}, \quad y(0)=2,\quad y'(0)=0\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{20}\)
\(y''+2y'+10y=e^{x/2}\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{21}\)
\(y''+6y'+10y=e^{-3x}\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{22}\)
Show that the method suggested for finding a particular solution in Exercises \((2.3E.16)\) to \((2.3E.21)\) won't yield a particular solution of
\begin{equation}\label{eq:2.3E.3}
y''-7y'+12y=5e^{4x};
\end{equation}
that is, \eqref{eq:2.3E.3} doesn't have a particular solution of the form \(y_p=Ae^{4x}\).
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{23}\)
Prove: If \(\alpha\) and \(M\) are constants and \(M\ne0\) then constant coefficient equation
\begin{eqnarray*}
ay''+by'+cy=M e^{\alpha x}
\end{eqnarray*}
has a particular solution \(y_p=Ae^{\alpha x}\) (\(A=\) constant) if and only if \(e^{\alpha x}\) isn't a solution of the complementary equation.
- Answer
-
Add texts here. Do not delete this text first.
If \(\omega\) is a constant, differentiating a linear combination of \(\cos\omega x\) and \(\sin\omega x\) with respect to \(x\) yields another linear combination of \(\cos\omega x\) and \(\sin\omega x\). In Exercises \((2.3E.24)\) to \((2.3E.29)\) use this to find a particular solution of the equation. Then find the general solution and, where indicated, solve the initial value problem and graph the solution.
Exercise \(\PageIndex{24}\)
\(y''-8y'+16y=23\cos x-7\sin x\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{25}\)
\(y''+y'=-8\cos2x+6\sin2x\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{26}\)
\(y''-2y'+3y=-6\cos3x+6\sin3x\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{27}\)
\(y''+6y'+13y=18\cos x+6\sin x \)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{28}\)
\(y''+7y'+12y=-2\cos2x+36\sin2x, \quad y(0)=-3,\quad y'(0)=3\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{29}\)
\(y''-6y'+9y=18\cos3x+18\sin3x, \quad y(0)=2,\quad y'(0)=2\)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{30}\)
Find the general solution of
\begin{eqnarray*}
y''+\omega_0^2y =M\cos\omega x+N\sin\omega x,
\end{eqnarray*}
where \(M\) and \(N\) are constants and \(\omega\) and \(\omega_0\) are distinct positive numbers.
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{31}\)
Show that the method suggested for finding a particular solution in Exercises \((2.3E.24)\) to \((2.3E.29)\) won't yield a particular solution of
\begin{equation}\label{eq:2.3E.4}
y''+y=\cos x+\sin x;
\end{equation}
that is, \eqref{eq:2.3E.4} does not have a particular solution of the form \(y_p=A\cos x+B\sin x\).
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{32}\)
Prove: If \(M\), \(N\) are constants (not both zero) and \(\omega>0\), the constant coefficient equation
\begin{equation}\label{eq:2.3E.5}
ay''+by'+cy=M\cos\omega x+N\sin\omega x
\end{equation}
has a particular solution that's a linear combination of \(\cos\omega x\) and \(\sin\omega x\) if and only if the left side of \eqref{eq:2.3E.5} is not of the form \(a(y''+\omega^2y)\), so that \(\cos\omega x\) and \(\sin\omega x\) are solutions of the complementary equation.
- Answer
-
Add texts here. Do not delete this text first.
In Exercises \((2.3E.33)\) to \(2.3E.38)\), refer to the cited exercises and use the principal of superposition to find a particular solution. Then find the general solution.
Exercise \(\PageIndex{33}\)
\(y''+5y'-6y=22+18x-18x^2+6e^{3x}\)
(See Exercises \((2.3E.1)\) and \((2.3E.16)\).)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{34}\)
\(y''-4y'+5y=1+5x+e^{2x}\)
(See Exercises \(2.3E.2)\) and \((2.3E.17)\).)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{35}\)
\(y''+8y'+7y=-8-x+24x^2+7x^3+10e^{-2x}\)
(See Exercises \((2.3E.3)\) and \((2.3E.18)\).)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{36}\)
\(y''-4y'+4y=2+8x-4x^2+e^{x}\)
(See Exercises \((2.3E.4)\) and \((2.3E.19)\).)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{37}\)
\(y''+2y'+10y=4+26x+6x^2+10x^3+e^{x/2}\)
(See Exercises \((2.3E.5)\) and \((2.3E.20)\).)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{38}\)
\(y''+6y'+10y=22+20x+e^{-3x}\)
(See Exercises \((2.3E.6)\) and \((2.3E.21)\).)
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{39}\)
Prove: If \(y_{p_1}\) is a particular solution of
\begin{eqnarray*}
P_0(x)y''+P_1(x)y'+P_2(x)y=F_1(x)
\end{eqnarray*}
on \((a,b)\) and \(y_{p_2}\) is a particular solution of
\begin{eqnarray*}
P_0(x)y''+P_1(x)y'+P_2(x)y=F_2(x)
\end{eqnarray*}
on \((a,b)\), then \(y_p=y_{p_1}+y_{p_2}\) is a solution of
\begin{eqnarray*}
P_0(x)y''+P_1(x)y'+P_2(x)y=F_1(x)+F_2(x)
\end{eqnarray*}
on \((a,b)\).
- Answer
-
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{40}\)
Suppose \(p\), \(q\), and \(f\) are continuous on \((a,b)\). Let \(y_1\), \(y_2\), and \(y_p\) be twice differentiable on \((a,b)\), such that \(y=c_1y_1+c_2y_2+y_p\) is a solution of
\begin{eqnarray*}
y''+p(x)y'+q(x)y=f
\end{eqnarray*}
on \((a,b)\) for every choice of the constants \(c_1,c_2\). Show that \(y_1\) and \(y_2\) are solutions of the complementary equation on \((a,b)\).
- Answer
-
Add texts here. Do not delete this text first.