5.1E: Homogeneous Linear Equations (Exercises)

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$trench-1995.jpg$
Contributed by William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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[exer:5.1.1]

Verify that $y_1=e^{2x}$ and $y_2=e^{5x}$ are solutions of

\[y''-7y'+10y=0 \eqno{\rm (A)}\nonumber \]

on $(-\infty,\infty)$.

Verify that if $c_1$ and $c_2$ are arbitrary constants then $y=c_1e^{2x}+c_2e^{5x}$ is a solution of (A) on $(-\infty,\infty)$.

Solve the initial value problem

\[y''-7y'+10y=0,\quad y(0)=-1,\quad y'(0)=1.\nonumber \]

Solve the initial value problem

\[y''-7y'+10y=0,\quad y(0)=k_0,\quad y'(0)=k_1.\nonumber \]

[exer:5.1.2]

Verify that $y_1=e^x\cos x$ and $y_2=e^x\sin x$ are solutions of

\[y''-2y'+2y=0 \eqno{\rm (A)}\nonumber \]

on $(-\infty,\infty)$.

Verify that if $c_1$ and $c_2$ are arbitrary constants then $y=c_1e^x\cos x+c_2e^x\sin x$ is a solution of (A) on $(-\infty,\infty)$.

Solve the initial value problem

\[y''-2y'+2y=0,\quad y(0)=3,\quad y'(0)=-2.\nonumber \]

Solve the initial value problem

\[y''-2y'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1.\nonumber \]

[exer:5.1.3]

Verify that $y_1=e^x$ and $y_2=xe^x$ are solutions of

\[y''-2y'+y=0 \eqno{\rm (A)}\nonumber \]

on $(-\infty,\infty)$.

Verify that if $c_1$ and $c_2$ are arbitrary constants then $y=e^x(c_1+c_2x)$ is a solution of (A) on $(-\infty,\infty)$.

Solve the initial value problem

\[y''-2y'+y=0,\quad y(0)=7,\quad y'(0)=4.\nonumber \]

Solve the initial value problem

\[y''-2y'+y=0,\quad y(0)=k_0,\quad y'(0)=k_1.\nonumber \]

[exer:5.1.4]

Verify that $y_1=1/(x-1)$ and $y_2=1/(x+1)$ are solutions of

\[(x^2-1)y''+4xy'+2y=0 \eqno{\rm (A)}\nonumber \]

on $(-\infty,-1)$, $(-1,1)$, and $(1,\infty)$. What is the general solution of (A) on each of these intervals?

Solve the initial value problem

\[(x^2-1)y''+4xy'+2y=0,\quad y(0)=-5,\quad y'(0)=1.\nonumber \]

What is the interval of validity of the solution?

Graph the solution of the initial value problem.

Verify Abel’s formula for $y_1$ and $y_2$, with $x_0=0$.

[exer:5.1.5] Compute the Wronskians of the given sets of functions.

(a) $\{1, e^x\}$

(b) $\{e^x, e^x \sin x\}$

(c) $\{x+1, x^2+2\}$

(d) $\{ x^{1/2}, x^{-1/3}\}$

(e) $\{ \frac{\sin x}{x}, \frac{\cos x}{x}\}$	(f) $\{ x \ln\|x\|, x^2\ln\|x\|\}$
(g) $\{e^x\cos\sqrt x, e^x\sin\sqrt x\}$

[exer:5.1.6] Find the Wronskian of a given set $\{y_1,y_2\}$ of solutions of

\[y''+3(x^2+1)y'-2y=0,\nonumber \]

given that $W(\pi)=0$.

[exer:5.1.7] Find the Wronskian of a given set $\{y_1,y_2\}$ of solutions of

\[(1-x^2)y''-2xy'+\alpha(\alpha+1)y=0,\nonumber \]

given that $W(0)=1$. (This is Legendre’s equation.)

[exer:5.1.8] Find the Wronskian of a given set $\{y_1,y_2\}$ of solutions of

\[x^2y''+xy'+(x^2-\nu^2)y=0 ,\nonumber \]

given that $W(1)=1$. (This is Bessel’s equation.)

[exer:5.1.9] (This exercise shows that if you know one nontrivial solution of $y''+p(x)y'+q(x)y=0$, you can use Abel’s formula to find another.)

Suppose $p$ and $q$ are continuous and $y_1$ is a solution of

\[y''+p(x)y'+q(x)y=0 \eqno{\rm (A)}\nonumber \]

that has no zeros on $(a,b)$. Let $P(x)=\int p(x)\,dx$ be any antiderivative of $p$ on $(a,b)$.

Show that if $K$ is an arbitrary nonzero constant and $y_2$ satisfies

\[y_1y_2'-y_1'y_2=Ke^{-P(x)} \eqno{\rm (B)}\nonumber \]

on $(a,b)$, then $y_2$ also satisfies (A) on $(a,b)$, and $\{y_1,y_2\}$ is a fundamental set of solutions on (A) on $(a,b)$.

Conclude from

a

that if $y_2=uy_1$ where $u'=K{e^{-P(x)}\over y_1^2(x)}$, then $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(a,b)$.

[exer:5.1.10] $y''-2y'-3y=0$; $y_1=e^{3x}$

[exer:5.1.11] $y''-6y'+9y=0$; $y_1=e^{3x}$

[exer:5.1.12] $y''-2ay'+a^2y=0$ ($a=$ constant); $y_1=e^{ax}$

[exer:5.1.13] $x^2y''+xy'-y=0$; $y_1=x$

[exer:5.1.14] $x^2y''-xy'+y=0$; $y_1=x$

[exer:5.1.15] $x^2y''-(2a-1)xy'+a^2y=0$ ($a=$ nonzero constant); $x>0$; $y_1=x^a$

[exer:5.1.16] $4x^2y''-4xy'+(3-16x^2)y=0$; $y_1=x^{1/2}e^{2x}$

[exer:5.1.17] $(x-1)y''-xy'+y=0$; $y_1=e^x$

[exer:5.1.18] $x^2y''-2xy'+(x^2+2)y=0$; $y_1=x\cos x$

[exer:5.1.19] $4x^2(\sin x)y''-4x(x\cos x+\sin x)y'+(2x\cos x+3\sin x)y=0$; $y_1=x^{1/2}$

[exer:5.1.20] $(3x-1)y''-(3x+2)y'-(6x-8)y=0$; $y_1=e^{2x}$

[exer:5.1.21] $(x^2-4)y''+4xy'+2y=0$; $y_1={1\over x-2}$

[exer:5.1.22] $(2x+1)xy''-2(2x^2-1)y'-4(x+1)y=0$;$y_1={1\over x}$

[exer:5.1.23] $(x^2-2x)y''+(2-x^2)y'+(2x-2)y=0$;$y_1=e^x$

[exer:5.1.24] Suppose $p$ and $q$ are continuous on an open interval $(a,b)$ and let $x_0$ be in $(a,b)$. Use Theorem [thmtype:5.1.1} to show that the only solution of the initial value problem

\[y''+p(x)y'+q(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0\nonumber \]

on $(a,b)$ is the trivial solution $y\equiv0$.

[exer:5.1.25] Suppose $P_0$, $P_1$, and $P_2$ are continuous on $(a,b)$ and let $x_0$ be in $(a,b)$. Show that if either of the following statements is true then $P_0(x)=0$ for some $x$ in $(a,b)$.

The initial value problem

\[P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \]

has more than one solution on $(a,b)$.

The initial value problem

\[P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0\nonumber \]

has a nontrivial solution on $(a,b)$.

[exer:5.1.26] Suppose $p$ and $q$ are continuous on $(a,b)$ and $y_1$ and $y_2$ are solutions of

\[y''+p(x)y'+q(x)y=0 \eqno{\rm (A)}\nonumber \]

on $(a,b)$. Let

\[z_1=\alpha y_1+\beta y_2\mbox{\quad and \quad} z_2=\gamma y_1+\delta y_2,\nonumber \]

where $\alpha$, $\beta$, $\gamma$, and $\delta$ are constants. Show that if $\{z_1,z_2\}$ is a fundamental set of solutions of (A) on $(a,b)$ then so is $\{y_1,y_2\}$.

[exer:5.1.27] Suppose $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a fundamental set of solutions of

\[y''+p(x)y'+q(x)y=0 \eqno{\rm (A)}\nonumber \]

on $(a,b)$. Let

\[z_1=\alpha y_1+\beta y_2\mbox{\quad and \quad} z_2=\gamma y_1+\delta y_2,\nonumber \]

where $\alpha,\beta,\gamma$, and $\delta$ are constants. Show that $\{z_1,z_2\}$ is a fundamental set of solutions of (A) on $(a,b)$ if and only if $\alpha\gamma-\beta\delta\ne0$.

[exer:5.1.28] Suppose $y_1$ is differentiable on an interval $(a,b)$ and $y_2=ky_1$, where $k$ is a constant. Show that the Wronskian of $\{y_1,y_2\}$ is identically zero on $(a,b)$.

[exer:5.1.29] Let

\[y_1=x^3\quad\mbox{ and }\quad y_2=\left\{\begin{array}{rl} x^3,&x\ge 0,\\ -x^3,&x<0.\end{array}\right.\nonumber \]

Show that the Wronskian of $\{y_1,y_2\}$ is defined and identically zero on $(-\infty,\infty)$.

Suppose $a<0<b$. Show that $\{y_1,y_2\}$ is linearly independent on $(a,b)$.

Use Exercise [exer:5.1.25}

b

to show that these results don’t contradict Theorem [thmtype:5.1.5} , because neither $y_1$ nor $y_2$ can be a solution of an equation

\[y''+p(x)y'+q(x)y=0\nonumber \]

on $(a,b)$ if $p$ and $q$ are continuous on $(a,b)$.

[exer:5.1.30] Suppose $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a set of solutions of

\[y''+p(x)y'+q(x)y=0\nonumber \]

on $(a,b)$ such that either $y_1(x_0)=y_2(x_0)=0$ or $y_1'(x_0)=y_2'(x_0)=0$ for some $x_0$ in $(a,b)$. Show that $\{y_1,y_2\}$ is linearly dependent on $(a,b)$.

[exer:5.1.31] Suppose $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a fundamental set of solutions of

\[y''+p(x)y'+q(x)y=0\nonumber \]

on $(a,b)$. Show that if $y_1(x_1)=y_1(x_2)=0$, where $a<x_1<x_2<b$, then $y_2(x)=0$ for some $x$ in $(x_1,x_2)$.

[exer:5.1.32] Suppose $p$ and $q$ are continuous on $(a,b)$ and every solution of

\[y''+p(x)y'+q(x)y=0 \eqno{\rm (A)}\nonumber \]

on $(a,b)$ can be written as a linear combination of the twice differentiable functions $\{y_1,y_2\}$. Use Theorem [thmtype:5.1.1} to show that $y_1$ and $y_2$ are themselves solutions of (A) on $(a,b)$.

[exer:5.1.33] Suppose $p_1$, $p_2$, $q_1$, and $q_2$ are continuous on $(a,b)$ and the equations

\[y''+p_1(x)y'+q_1(x)y=0 \mbox{\quad and \quad} y''+p_2(x)y'+q_2(x)y=0\nonumber \]

have the same solutions on $(a,b)$. Show that $p_1=p_2$ and $q_1=q_2$ on $(a,b)$.

[exer:5.1.34] (For this exercise you have to know about $3\times 3$ determinants.) Show that if $y_1$ and $y_2$ are twice continuously differentiable on $(a,b)$ and the Wronskian $W$ of $\{y_1,y_2\}$ has no zeros in $(a,b)$ then the equation

\[\frac{1}{W} \left| \begin{array}{ccc} y & y_1 & y_2 \

\[2\jot] y' & y'_1 & y'_2 \

\[2\jot] y'' & y_1'' & y_2'' \end{array} \right|=0\nonumber \]

can be written as

\[y''+p(x)y'+q(x)y=0, \eqno{\rm (A)}\nonumber \]

where $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(a,b)$.

[exer:5.1.35] Use the method suggested by Exercise [exer:5.1.34} to find a linear homogeneous equation for which the given functions form a fundamental set of solutions on some interval.

(a) $e^x \cos 2x, \quad e^x \sin 2x$

(b) $x, \quad e^{2x}$

(c) $x, \quad x \ln x$

(d) $\cos (\ln x), \quad \sin (\ln x)$

(e) $\cosh x, \quad \sinh x$

(f) $x^2-1, \quad x^2+1$

[exer:5.1.36] Suppose $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a fundamental set of solutions of

\[y''+p(x)y'+q(x)y=0 \eqno{\rm (A)}\nonumber \]

on $(a,b)$. Show that if $y$ is a solution of (A) on $(a,b)$, there’s exactly one way to choose $c_1$ and $c_2$ so that $y=c_1y_1+c_2y_2$ on $(a,b)$.

[exer:5.1.37] Suppose $p$ and $q$ are continuous on $(a,b)$ and $x_0$ is in $(a,b)$. Let $y_1$ and $y_2$ be the solutions of

\[y''+p(x)y'+q(x)y=0 \eqno{\rm (A)}\nonumber \]

such that

\[y_1(x_0)=1, \quad y'_1(x_0)=0\mbox{\quad and \quad} y_2(x_0)=0,\; y'_2(x_0)=1.\nonumber \]

(Theorem [thmtype:5.1.1} implies that each of these initial value problems has a unique solution on $(a,b)$.)

Show that $\{y_1,y_2\}$ is linearly independent on $(a,b)$.

Show that an arbitrary solution $y$ of (A) on $(a,b)$ can be written as $y=y(x_0)y_1+y'(x_0)y_2$.

Express the solution of the initial value problem

\[y''+p(x)y'+q(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \]

as a linear combination of $y_1$ and $y_2$.

[exer:5.1.38] Find solutions $y_1$ and $y_2$ of the equation $y''=0$ that satisfy the initial conditions

\[y_1(x_0)=1, \quad y'_1(x_0)=0 \mbox{\quad and \quad} y_2(x_0)=0, \quad y'_2(x_0)=1.\nonumber \]

Then use Exercise [exer:5.1.37} (c) to write the solution of the initial value problem

\[y''=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]

as a linear combination of $y_1$ and $y_2$.

[exer:5.1.39] Let $x_0$ be an arbitrary real number. Given (Example

Example $\PageIndex{1}$:

Add text here. For the automatic number to work, you need to add the “AutoNum” template (preferably at5.1.1} ) that $e^x$ and $e^{-x}$ are solutions of $y''-y=0$, find solutions $y_1$ and $y_2$ of $y''-y=0$ such that

\[y_1(x_0)=1, \quad y'_1(x_0)=0\mbox{\quad and \quad} y_2(x_0)=0,\; y'_2(x_0)=1.\nonumber \]

Then use Exercise [exer:5.1.37} (c) to write the solution of the initial value problem

\[y''-y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \]

as a linear combination of $y_1$ and $y_2$.

[exer:5.1.40] Let $x_0$ be an arbitrary real number. Given (Example

Example $\PageIndex{1}$:

Add text here. For the automatic number to work, you need to add the “AutoNum” template (preferably at5.1.2} ) that $\cos\omega x$ and $\sin\omega x$ are solutions of $y''+\omega^2y=0$, find solutions of $y''+\omega^2y=0$ such that

\[y_1(x_0)=1, \quad y'_1(x_0)=0\mbox{\quad and \quad} y_2(x_0)=0,\; y'_2(x_0)=1.\nonumber \]

Then use Exercise [exer:5.1.37} (c) to write the solution of the initial value problem

\[y''+\omega^2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \]

as a linear combination of $y_1$ and $y_2$. Use the identities

\[\begin{aligned} \cos(A+B)&=&\cos A\cos B-\sin A\sin B\\ \sin(A+B)&=&\sin A\cos B+\cos A\sin B\end{aligned}\nonumber \]

to simplify your expressions for $y_1$, $y_2$, and $y$.

[exer:5.1.41] Recall from Exercise [exer:5.1.4} that $1/(x-1)$ and $1/(x+1)$ are solutions of

\[(x^2-1)y''+4xy'+2y=0 \eqno{\rm (A)}\nonumber \]

on $(-1,1)$. Find solutions of (A) such that

\[y_1(0)=1, \quad y'_1(0)=0\mbox{\quad and \quad} y_2(0)=0,\; y'_2(0)=1.\nonumber \]

Then use Exercise [exer:5.1.37} (c) to write the solution of initial value problem

\[(x^2-1)y''+4xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]

as a linear combination of $y_1$ and $y_2$.

[exer:5.1.42]

Verify that $y_1=x^2$ and $y_2=x^3$ satisfy

\[x^2y''-4xy'+6y=0 \eqno{\rm (A)}\nonumber \]

on $(-\infty,\infty)$ and that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(-\infty,0)$ and $(0,\infty)$.

Let $a_1$, $a_2$, $b_1$, and $b_2$ be constants. Show that

\[y=\left\{\begin{array}{rr} a_1x^2+a_2x^3,&x\ge 0,\\ b_1x^2+b_2x^3,&x<0\phantom{,} \end{array}\right.\nonumber \]

is a solution of (A) on $(-\infty,\infty)$ if and only if $a_1=b_1$. From this, justify the statement that $y$ is a solution of (A) on $(-\infty,\infty)$ if and only if

\[y=\left\{\begin{array}{rr} c_1x^2+c_2x^3,&x\ge 0,\\ c_1x^2+c_3x^3,&x<0, \end{array}\right.\nonumber \]

where $c_1$, $c_2$, and $c_3$ are arbitrary constants.

For what values of $k_0$ and $k_1$ does the initial value problem

\[x^2y''-4xy'+6y=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]

have a solution? What are the solutions?

Show that if $x_0\ne0$ and $k_0,k_1$ are arbitrary constants, the initial value problem

\[x^2y''-4xy'+6y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \eqno{\rm (B)}\nonumber \]

has infinitely many solutions on $(-\infty,\infty)$. On what interval does (B) have a unique solution?

[exer:5.1.43]

Verify that $y_1=x$ and $y_2=x^2$ satisfy

\[x^2y''-2xy'+2y=0 \eqno{\rm (A)}\nonumber \]

on $(-\infty,\infty)$ and that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(-\infty,0)$ and $(0,\infty)$.

Let $a_1$, $a_2$, $b_1$, and $b_2$ be constants. Show that

\[y=\left\{\begin{array}{rr} a_1x+a_2x^2,&x\ge 0,\\ b_1x+b_2x^2,&x<0\phantom{,} \end{array}\right.\nonumber \]

is a solution of (A) on $(-\infty,\infty)$ if and only if $a_1=b_1$ and $a_2=b_2$. From this, justify the statement that the general solution of (A) on $(-\infty,\infty)$ is $y=c_1x+c_2x^2$, where $c_1$ and $c_2$ are arbitrary constants.

For what values of $k_0$ and $k_1$ does the initial value problem

\[x^2y''-2xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]

have a solution? What are the solutions?

Show that if $x_0\ne0$ and $k_0,k_1$ are arbitrary constants then the initial value problem

\[x^2y''-2xy'+2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \]

has a unique solution on $(-\infty,\infty)$.

[exer:5.1.44]

Verify that $y_1=x^3$ and $y_2=x^4$ satisfy

\[x^2y''-6xy'+12y=0 \eqno{\rm (A)}\nonumber \]

on $(-\infty,\infty)$, and that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(-\infty,0)$ and $(0,\infty)$.

Show that $y$ is a solution of (A) on $(-\infty,\infty)$ if and only if

\[y=\left\{\begin{array}{rr} a_1x^3+a_2x^4,&x\ge 0,\\ b_1x^3+b_2x^4,&x<0, \end{array}\right.\nonumber \]

where $a_1$, $a_2$, $b_1$, and $b_2$ are arbitrary constants.

For what values of $k_0$ and $k_1$ does the initial value problem

\[x^2y''-6xy'+12y=0, \quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]

have a solution? What are the solutions?

Show that if $x_0\ne0$ and $k_0,k_1$ are arbitrary constants then the initial value problem

\[x^2y''-6xy'+12y=0, \quad y(x_0)=k_0,\quad y'(x_0)=k_1 \eqno{\rm (B)}\nonumber \]

has infinitely many solutions on $(-\infty,\infty)$. On what interval does (B) have a unique solution?

(e) \(\{ \frac{\sin x}{x}, \frac{\cos x}{x}\}\)	(f) \(\{ x \ln\|x\|, x^2\ln\|x\|\}\)
(g) \(\{e^x\cos\sqrt x, e^x\sin\sqrt x\}\)