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# 2.4E: Exercises

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In Exercises $$(2.4E.1)$$ to $$(2.4E.14)$$, find a particular solution.

## Exercise $$\PageIndex{1}$$

$$y''-3y'+2y=e^{3x}(1+x)$$

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## Exercise $$\PageIndex{2}$$

$$y''-6y'+5y=e^{-3x}(35-8x)$$

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## Exercise $$\PageIndex{3}$$

$$y''-2y'-3y=e^x(-8+3x)$$

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## Exercise $$\PageIndex{4}$$

$$y''+2y'+y=e^{2x}(-7-15x+9x^2)$$

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## Exercise $$\PageIndex{5}$$

$$y''+4y=e^{-x}(7-4x+5x^2)$$

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## Exercise $$\PageIndex{6}$$

$$y''-y'-2y=e^x(9+2x-4x^2)$$

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## Exercise $$\PageIndex{7}$$

$$y''-4y'-5y=-6xe^{-x}$$

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## Exercise $$\PageIndex{8}$$

$$y''-3y'+2y=e^x(3-4x)$$

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## Exercise $$\PageIndex{9}$$

$$y''+y'-12y=e^{3x}(-6+7x)$$

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## Exercise $$\PageIndex{10}$$

$$2y''-3y'-2y=e^{2x}(-6+10x)$$

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## Exercise $$\PageIndex{11}$$

$$y''+2y'+y=e^{-x}(2+3x)$$

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## Exercise $$\PageIndex{12}$$

$$y''-2y'+y=e^x(1-6x)$$

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## Exercise $$\PageIndex{13}$$

$$y''-4y'+4y=e^{2x}(1-3x+6x^2)$$

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## Exercise $$\PageIndex{14}$$

$$9y''+6y'+y=e^{-x/3}(2-4x+4x^2)$$

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In Exercises $$(2.4E.15)$$ to $$(2.4E.19)$$, find the general solution.

## Exercise $$\PageIndex{15}$$

$$y''-3y'+2y=e^{3x}(1+x)$$

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## Exercise $$\PageIndex{16}$$

$$y''-6y'+8y=e^x(11-6x)$$

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## Exercise $$\PageIndex{17}$$

$$y''+6y'+9y=e^{2x}(3-5x)$$

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## Exercise $$\PageIndex{18}$$

$$y''+2y'-3y=-16xe^x$$

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## Exercise $$\PageIndex{19}$$

$$y''-2y'+y=e^x(2-12x)$$

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In Exercises $$(2.4E.20)$$ to $$(2.4E.23)$$, solve the initial value problem and plot the solution.

## Exercise $$\PageIndex{20}$$

$$y''-4y'-5y=9e^{2x}(1+x), \quad y(0)=0,\quad y'(0)=-10$$

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## Exercise $$\PageIndex{21}$$

$$y''+3y'-4y=e^{2x}(7+6x), \quad y(0)=2,\quad y'(0)=8$$

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## Exercise $$\PageIndex{22}$$

$$y''+4y'+3y=-e^{-x}(2+8x), \quad y(0)=1,\quad y'(0)=2$$

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## Exercise $$\PageIndex{23}$$

$$y''-3y'-10y=7e^{-2x}, \quad y(0)=1,\quad y'(0)=-17$$

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In Exercises $$(2.4E.24)$$ to $$(2.4E.29)$$, use the principle of superposition to find a particular solution.

## Exercise $$\PageIndex{24}$$

$$y''+y'+y=xe^x+e^{-x}(1+2x)$$

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## Exercise $$\PageIndex{25}$$

$$y''-7y'+12y=-e^x(17-42x)-e^{3x}$$

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## Exercise $$\PageIndex{26}$$

$$y''-8y'+16y=6xe^{4x}+2+16x+16x^2$$

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## Exercise $$\PageIndex{27}$$

$$y''-3y'+2y=-e^{2x}(3+4x)-e^x$$

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## Exercise $$\PageIndex{28}$$

$$y''-2y'+2y=e^x(1+x)+e^{-x}(2-8x+5x^2)$$

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## Exercise $$\PageIndex{29}$$

$$y''+y=e^{-x}(2-4x+2x^2)+e^{3x}(8-12x-10x^2)$$

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## Exercise $$\PageIndex{30}$$

(a) Prove that $$y$$ is a solution of the constant coefficient equation

\label{eq:2.4E.1}
ay''+by'+cy=e^{\alpha x}G(x)

if and only if $$y=ue^{\alpha x}$$, where $$u$$ satisfies

\label{eq:2.4E.2}
au''+p'(\alpha)u'+p(\alpha)u=G(x)

and $$p(r)=ar^2+br+c$$ is the characteristic polynomial of the complementary equation

\begin{eqnarray*}
ay''+by'+cy=0.
\end{eqnarray*}

For the rest of this exercise, let $$G$$ be a polynomial. Give the requested proofs for the case where

\begin{eqnarray*}
G(x)=g_0+g_1x+g_2x^2+g_3x^3.
\end{eqnarray*}

(b) Prove that if $$e^{\alpha x}$$ isn't a solution of the complementary equation then \eqref{eq:2.4E.2} has a particular solution of the form $$u_p=A(x)$$, where $$A$$ is a polynomial of the same degree as $$G$$, as in Example $$(2.4.4)$$. Conclude that \eqref{eq:2.4E.1} has a particular solution of the form $$y_p=e^{\alpha x}A(x)$$.

(c) Show that if $$e^{\alpha x}$$ is a solution of the complementary equation and $$xe^{\alpha x}$$ isn't, then \eqref{eq:2.4E.2} has a particular solution of the form $$u_p=xA(x)$$, where $$A$$ is a polynomial of the same degree as $$G$$, as in Example $$(2.4.5)$$. Conclude that \eqref{eq:2.4E.1} has a particular solution of the form $$y_p=xe^{\alpha x}A(x)$$.

(d) Show that if $$e^{\alpha x}$$ and $$xe^{\alpha x}$$ are both solutions of the complementary equation then \eqref{eq:2.4E.2} has a particular solution of the form $$u_p=x^2A(x)$$, where $$A$$ is a polynomial of the same degree as $$G$$, and $$x^2A(x)$$ can be obtained by integrating $$G/a$$ twice, taking the constants of integration to be zero, as in Example $$(2.4.6)$$. Conclude that \eqref{eq:2.4E.1} has a particular solution of the form $$y_p=x^2e^{\alpha x}A(x)$$.

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Exercises $$(2.4E.31)$$ to $$(2.4E.36)$$ treat the equations considered in Examples $$(2.4.1)$$ to $$(2.4.6)$$. Substitute the suggested form of $$y_p$$ into the equation and equate the resulting coefficients of like functions on the two sides of the resulting equation to derive a set of simultaneous equations for the coefficients in $$y_p$$. Then solve for the coefficients to obtain $$y_p$$. Compare the work you've done with the work required to obtain the same results in Examples $$(2.4.1)$$ to $$(2.4.6)$$.

## Exercise $$\PageIndex{31}$$

Compare with Example $$(2.4.1)$$:

\begin{eqnarray*}
\end{eqnarray*}

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## Exercise $$\PageIndex{32}$$

Compare with Example $$(2.4.2)$$:

\begin{eqnarray*}
\end{eqnarray*}

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## Exercise $$\PageIndex{33}$$

Compare with Example $$(2.4.3)$$:

\begin{eqnarray*}
\end{eqnarray*}

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## Exercise $$\PageIndex{34}$$

Compare with Example $$(2.4.4)$$:

\begin{eqnarray*}
\end{eqnarray*}

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## Exercise $$\PageIndex{35}$$

Compare with Example $$(2.4.5)$$:

\begin{eqnarray*}
\end{eqnarray*}

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## Exercise $$\PageIndex{36}$$

Compare with Example $$(2.4.6)$$:

\begin{eqnarray*}
\end{eqnarray*}

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## Exercise $$\PageIndex{37}$$

Write $$y=ue^{\alpha x}$$ to find the general solution.

(a) $$y''+2y'+y=\displaystyle{e^{-x}\over\sqrt x}$$

(b) $$y''+6y'+9y=e^{-3x}\ln x$$

(c) $$y''-4y'+4y=\displaystyle{e^{2x}\over1+x}$$

(d) $$4y''+4y'+y=\displaystyle{4e^{-x/2}\left({1\over x}+x\right)}$$

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## Exercise $$\PageIndex{38}$$

Suppose $$\alpha\ne0$$ and $$k$$ is a positive integer. In most calculus books integrals like $$\int x^k e^{\alpha x}\,dx$$ are evaluated by integrating by parts $$k$$ times. This exercise presents another method. Let

\begin{eqnarray*}
y=\int e^{\alpha x}P(x)\,dx
\end{eqnarray*}

with

\begin{eqnarray*}
P(x)=p_0+p_1x+\cdots+p_kx^k, \mbox{\quad (where $p_k\ne0$)}.
\end{eqnarray*}

(a) Show that $$y=e^{\alpha x}u$$, where

\label{eq:2.4E.3}
u'+\alpha u=P(x).

(b) Show that \eqref{eq:2.4E.3} has a particular solution of the form

\begin{eqnarray*}
u_p=A_0+A_1x+\cdots+A_kx^k,
\end{eqnarray*}

where $$A_k$$, $$A_{k-1}$$, $$\dots$$, $$A_0$$ can be computed successively by equating coefficients of $$x^k,x^{k-1}, \dots,1$$ on both sides of the equation

\begin{eqnarray*}
u_p'+\alpha u_p=P(x).
\end{eqnarray*}

(c) Conclude that

\begin{eqnarray*}
\int e^{\alpha x}P(x)\,dx=\left(A_0+A_1x+\cdots+A_kx^k\right)e^{\alpha x} +c,
\end{eqnarray*}

where $$c$$ is a constant of integration.

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## Exercise $$\PageIndex{39}$$

Use the method of Exercise $$(2.4E.38)$$ to evaluate the integral.

(a) $$\int e^x(4+x)\,dx$$

(b) $$\int e^{-x}(-1+x^2)\,dx$$

(c) $$\int x^3e^{-2x}\,dx$$

(d) $$\int e^x(1+x)^2\,dx$$

(e) $$\int e^{3x}(-14+30x+27x^2)\,dx$$

(f) $$\int e^{-x}(1+6x^2-14x^3+3x^4)\,dx$$

## Exercise $$\PageIndex{40}$$
Use the method suggested in Exercise $$(2.4E.38)$$ to evaluate $$\int x^ke^{\alpha x}\,dx$$, where $$k$$ is an arbitrary positive integer and $$\alpha\ne0$$.