
# 5.1E: Homogeneous Linear Equations (Exercises)

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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?