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2nd Order Homogeneous DE Exercises


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1.1

(a)

Verify that {$y_1=e^{2x}$} and {$y_2=e^{5x}$} are solutions of {$$ y''-7y'+10y=0 \quad\quad \text{(A)}$$} on {$(-\infty,\infty)$}

(b)

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 {$(-\infty,\infty)$}.

(c)

Solve the initial value problem {$$ y''-7y'+10y=0,\quad y(0)=-1,\quad y'(0)=1. $$}

(d)

Solve the initial value problem {$$ y''-7y'+10y=0,\quad y(0)=k_0,\quad y'(0)=k_1. $$}

\item\label{exer:5.1.2} \begin{alist} \item % (a) Verify that $y_1=e^x\cos x$ and $y_2=e^x\sin x$ are solutions of $$ y''-2y'+2y=0 \eqno{\rm (A)} $$ on $(-\infty,\infty)$. \item % (b) 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 $(-\infty,\infty)$. \item % (c) Solve the initial value problem $$ y''-2y'+2y=0,\quad y(0)=3,\quad y'(0)=-2. $$ \item % (d) Solve the initial value problem $$ y''-2y'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1. $$ \end{alist}

\item\label{exer:5.1.3} \begin{alist} \item % (a) Verify that $y_1=e^x$ and $y_2=xe^x$ are solutions of $$ y''-2y'+y=0 \eqno{\rm (A)} $$ on $(-\infty,\infty)$. \item % (b) 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 $(-\infty,\infty)$. \item % (c) Solve the initial value problem $$ y''-2y'+y=0,\quad y(0)=7,\quad y'(0)=4. $$ \item % (d) Solve the initial value problem $$ y''-2y'+y=0,\quad y(0)=k_0,\quad y'(0)=k_1. $$ \end{alist}

\item\label{exer:5.1.4} \begin{alist} \item % (a) Verify that $y_1=1/(x-1)$ and $y_2=1/(x+1)$ are solutions of $$ (x^2-1)y''+4xy'+2y=0 \eqno{\rm (A)} $$ on $(-\infty,-1)$, $(-1,1)$, and $(1,\infty)$. What is the general solution of (A) on each of these intervals? \item % (b) Solve the initial value problem $$ (x^2-1)y''+4xy'+2y=0,\quad y(0)=-5,\quad y'(0)=1. $$ What is the interval of validity of the solution? \item % (c) \CGex Graph the solution of the initial value problem. \item % (d) Verify Abel's formula for $y_1$ and $y_2$, with $x_0=0$. \end{alist}

\item\label{exer:5.1.5} Compute the Wronskians of the given sets of functions.

\begin{tabular}[t]{@{}p{168pt}@{}p{168pt}} {\bf (a)} $\{1, e^x\}$ & {\bf (b)} $\{e^x, e^x \sin x\}$ \end{tabular}

\begin{tabular}[t]{@{}p{168pt}@{}p{168pt}} {\bf (c)} $\{x+1, x^2+2\}$ & {\bf (d)} $\{ x^{1/2}, x^{-1/3}\}$ \end{tabular}

\begin{tabular}[t]{@{}p{168pt}@{}p{168pt}} {\bf (e)} $\{\dst \frac{\sin x}{x}, \frac{\cos x}{x}\}$ & {\bf (f)} $\{ x \ln|x|, x^2\ln|x|\}$
{\bf (g)}

 $\{e^x\cos\sqrt x, e^x\sin\sqrt x\}$ &

\end{tabular}

\item\label{exer:5.1.6} Find the Wronskian of a given set $\{y_1,y_2\}$ of solutions of $$ y''+3(x^2+1)y'-2y=0, $$ given that $W(\pi)=0$.

\item\label{exer:5.1.7} Find the Wronskian of a given set $\{y_1,y_2\}$ of solutions of $$ (1-x^2)y''-2xy'+\alpha(\alpha+1)y=0, $$ given that $W(0)=1$. (This is \href{http://www-history.mcs.st-and.ac.uk/PictDisplay/Legendre.html(approve sites)} {\color{blue}\it Legendre's equation}.)

\item\label{exer:5.1.8} Find the Wronskian of a given set $\{y_1,y_2\}$ of solutions of $$ x^2y''+xy'+(x^2-\nu^2)y=0 , $$ given that $W(1)=1$. (This is \href{http://www-history.mcs.st-and.ac.uk/Mathematicians/Bessel.html(approve sites)} {\color{blue}\it Bessel's equation}.)

\item\label{exer:5.1.9} %\exerciseabel (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 $$ y''+p(x)y'+q(x)y=0 \eqno{\rm (A)} $$ that has no zeros on $(a,b)$. Let $P(x)=\int p(x)\,dx$ be any antiderivative of $p$ on $(a,b)$. \begin{alist} \item % (a) Show that if $K$ is an arbitrary nonzero constant and $y_2$ satisfies $$ y_1y_2'-y_1'y_2=Ke^{-P(x)} \eqno{\rm (B)} $$ 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 $(a,b)$.

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

\exercisetext{In Exercises~\ref{exer:5.1.10}--\ref{exer:5.1.23} use the method suggested by Exercise~\ref{exer:5.1.9} to find a second solution $y_2$ that isn't a constant multiple of the solution $y_1$. Choose $K$ conveniently to simplify $y_2$.}

\item\label{exer:5.1.10} $y''-2y'-3y=0$; \quad $y_1=e^{3x}$

\item\label{exer:5.1.11} $y''-6y'+9y=0$; \quad $y_1=e^{3x}$

\item\label{exer:5.1.12}% $y''-2ay'+a^2y=0$\; ($a=$ constant); \quad $y_1=e^{ax}$

\item\label{exer:5.1.13} $x^2y''+xy'-y=0$; \quad $y_1=x$

\item\label{exer:5.1.14} $x^2y''-xy'+y=0$; \quad $y_1=x$

\item\label{exer:5.1.15} $x^2y''-(2a-1)xy'+a^2y=0$\; ($a=$ nonzero constant);\, $x>0$; \quad $y_1=x^a$

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

\item\label{exer:5.1.17} $(x-1)y''-xy'+y=0$; \quad $y_1=e^x$

\item\label{exer:5.1.18} $x^2y''-2xy'+(x^2+2)y=0$; \quad $y_1=x\cos x$

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

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

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

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

\item\label{exer:5.1.23} $(x^2-2x)y''+(2-x^2)y'+(2x-2)y=0$;\quad $y_1=e^x$

\item\label{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~\ref{thmtype:5.1.1} to show that

 the only solution of the initial value problem

$$ y''+p(x)y'+q(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0 $$ on $(a,b)$ is the trivial solution $y\equiv0$.

\item\label{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 $(a,b)$. \begin{alist} \item % (a) The initial value problem $$ P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 $$ has more than one solution on $(a,b)$. \item % (b) The initial value problem $$ P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0 $$ has a nontrivial solution on $(a,b)$. \end{alist}

\item\label{exer:5.1.26} Suppose $p$ and $q$ are continuous on $(a,b)$ and

 $y_1$ and $y_2$ are solutions of

$$ y''+p(x)y'+q(x)y=0 \eqno{\rm (A)} $$ on $(a,b)$. Let $$ z_1=\alpha y_1+\beta y_2\mbox{\quad and \quad} z_2=\gamma y_1+\delta y_2, $$ 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 $\{y_1,y_2\}$.

\item\label{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 $$ y''+p(x)y'+q(x)y=0 \eqno{\rm (A)} $$ on $(a,b)$. Let $$ z_1=\alpha y_1+\beta y_2\mbox{\quad and \quad} z_2=\gamma y_1+\delta y_2, $$ 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$.

\item\label{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 $(a,b)$.

\item\label{exer:5.1.29} Let $$ y_1=x^3\quad\mbox{ and }\quad y_2=\left\{\begin{array}{rl} x^3,&x\ge 0,\\ -x^3,&x<0.\end{array}\right. $$ \begin{alist} \item % (a) Show that the Wronskian of $\{y_1,y_2\}$ is defined and identically zero on $(-\infty,\infty)$. \item % (b) Suppose $a<0<b$. Show that $\{y_1,y_2\}$ is linearly independent on $(a,b)$. \item % (b) Use Exercise~\ref{exer:5.1.25}\part{b} to show that these results don't contradict Theorem~\ref{thmtype:5.1.5}, because neither $y_1$ nor $y_2$ can be a solution of an equation $$ y''+p(x)y'+q(x)y=0 $$ on $(a,b)$ if $p$ and $q$ are continuous on $(a,b)$. \end{alist}

\item\label{exer:5.1.30} Suppose $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a set of solutions of $$ y''+p(x)y'+q(x)y=0 $$ 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 $(a,b)$.

\item\label{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 $$ y''+p(x)y'+q(x)y=0 $$ 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 $(x_1,x_2)$. \hint{Show that if $y_2$ has no zeros in $(x_1,x_2)$, then $y_1/y_2$ is either strictly increasing or strictly decreasing on $(x_1,x_2)$, and deduce a contradiction.}

\item\label{exer:5.1.32} Suppose $p$ and $q$ are continuous on $(a,b)$ and

 every solution of

$$ y''+p(x)y'+q(x)y=0 \eqno{\rm (A)} $$ on $(a,b)$ can be written as a linear combination of the twice differentiable functions $\{y_1,y_2\}$. Use Theorem~\ref{thmtype:5.1.1} to show that $y_1$ and $y_2$ are themselves solutions of (A) on $(a,b)$.

\item\label{exer:5.1.33} Suppose $p_1$, $p_2$, $q_1$, and $q_2$ are continuous on $(a,b)$ and the equations $$ y''+p_1(x)y'+q_1(x)y=0 \mbox{\quad and \quad} y''+p_2(x)y'+q_2(x)y=0 $$ have the same solutions on $(a,b)$. Show that $p_1=p_2$ and $q_1=q_2$ on $(a,b)$. \hint{Use Abel's formula.}

\item\label{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 $$ \frac{1}{W} \left| \begin{array}{ccc} y & y_1 & y_2 \\[2\jot] y' & y'_1 & y'_2 \\[2\jot] y & y_1 & y_2'' \end{array} \right|=0 $$ can be written as $$ y''+p(x)y'+q(x)y=0, \eqno{\rm (A)} $$ where $p$ and $q$ are continuous on $(a,b)$ and $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(a,b)$. \hint{Expand the determinant by cofactors of its first column.}

\item\label{exer:5.1.35} Use the method suggested by Exercise~\ref{exer:5.1.34} to find a linear homogeneous equation for which the given functions form a fundamental set of solutions on some interval.

\begin{tabular}[t]{@{}p{168pt}@{}p{168pt}} {\bf (a)} $e^x \cos 2x, \quad e^x \sin 2x$ & {\bf (b)} $x, \quad e^{2x}$ \end{tabular}

\begin{tabular}[t]{@{}p{168pt}@{}p{168pt}} {\bf (c)} $x, \quad x \ln x$ & {\bf (d)} $\cos (\ln x), \quad \sin (\ln x)$ \end{tabular}

\begin{tabular}[t]{@{}p{168pt}@{}p{168pt}} {\bf (e)} $\cosh x, \quad \sinh x$ & {\bf (f)} $ x^2-1, \quad x^2+1$ \end{tabular}

\item\label{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 $$ y''+p(x)y'+q(x)y=0 \eqno{\rm (A)} $$ 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 $(a,b)$.

\item\label{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 $$ y''+p(x)y'+q(x)y=0 \eqno{\rm (A)} $$

 such that

$$ y_1(x_0)=1, \quad y'_1(x_0)=0\mbox{\quad and \quad} y_2(x_0)=0,\; y'_2(x_0)=1. $$ (Theorem~\ref{thmtype:5.1.1} implies that each of these initial value problems has a unique solution on $(a,b)$.) \begin{alist} \item % (a) Show that $\{y_1,y_2\}$ is linearly independent on $(a,b)$. \item % (b) Show that an arbitrary solution $y$ of (A) on $(a,b)$ can be written as $y=y(x_0)y_1+y'(x_0)y_2$. \item % (c) Express the solution of the initial value problem $$ y''+p(x)y'+q(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 $$ as a linear combination of $y_1$ and $y_2$. \end{alist}

\item\label{exer:5.1.38} Find solutions $y_1$ and $y_2$ of the equation $y''=0$ that satisfy the initial conditions $$ y_1(x_0)=1, \quad y'_1(x_0)=0 \mbox{\quad and \quad} y_2(x_0)=0, \quad y'_2(x_0)=1. $$ Then use Exercise~\ref{exer:5.1.37}~{\bf (c)} to write the solution of the initial value problem $$ y''=0,\quad y(0)=k_0,\quad y'(0)=k_1 $$ as a linear combination of $y_1$ and $y_2$.

\item\label{exer:5.1.39} Let $x_0$ be an arbitrary real number. Given (Example~\ref{example:5.1.1}) 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 $$ y_1(x_0)=1, \quad y'_1(x_0)=0\mbox{\quad and \quad} y_2(x_0)=0,\; y'_2(x_0)=1. $$ Then use Exercise~\ref{exer:5.1.37}~{\bf (c)} to write the solution of the initial value problem $$ y''-y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 $$ as a linear combination of $y_1$ and $y_2$.

\item\label{exer:5.1.40} Let $x_0$ be an arbitrary real number. Given (Example~\ref{example:5.1.2}) that $\cos\omega x$ and $\sin\omega x$ are solutions of $y''+\omega^2y=0$, find solutions of

 $y''+\omega^2y=0$ such that

$$ y_1(x_0)=1, \quad y'_1(x_0)=0\mbox{\quad and \quad} y_2(x_0)=0,\; y'_2(x_0)=1. $$ Then use Exercise~\ref{exer:5.1.37}~{\bf (c)} to write the solution of the initial value problem $$ y''+\omega^2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 $$ as a linear combination of $y_1$ and $y_2$. Use the identities \begin{eqnarray*} \cos(A+B)&=&\cos A\cos B-\sin A\sin B
\sin(A+B)&=&\sin A\cos B+\cos A\sin B \end{eqnarray*} to simplify your expressions for $y_1$, $y_2$, and $y$.

\item\label{exer:5.1.41} Recall from Exercise~\ref{exer:5.1.4} that

 $1/(x-1)$ and $1/(x+1)$ are solutions of

$$ (x^2-1)y''+4xy'+2y=0 \eqno{\rm (A)} $$ on $(-1,1)$. Find solutions of (A)

 such that

$$ y_1(0)=1, \quad y'_1(0)=0\mbox{\quad and \quad} y_2(0)=0,\; y'_2(0)=1. $$ Then use Exercise~\ref{exer:5.1.37}~{\bf (c)} to write the solution of initial value problem $$ (x^2-1)y''+4xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1 $$ as a linear combination of $y_1$ and $y_2$.

\item\label{exer:5.1.42} \begin{alist} \item % (a) Verify that $y_1=x^2$ and $y_2=x^3$ satisfy $$ x^2y''-4xy'+6y=0 \eqno{\rm (A)} $$ on $(-\infty,\infty)$ and that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(-\infty,0)$ and $(0,\infty)$. \item % (b) Let $a_1$, $a_2$, $b_1$, and $b_2$ be constants. Show that $$ y=\left\{\begin{array}{rr} a_1x^2+a_2x^3,&x\ge 0,
b_1x^2+b_2x^3,&x<0\phantom{,} \end{array}\right. $$ 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 $$ y=\left\{\begin{array}{rr} c_1x^2+c_2x^3,&x\ge 0,
c_1x^2+c_3x^3,&x<0, \end{array}\right. $$ where $c_1$, $c_2$, and $c_3$ are arbitrary constants.

\item % (c)

 For what values of $k_0$ and  $k_1$

does the initial value problem $$ x^2y''-4xy'+6y=0,\quad y(0)=k_0,\quad y'(0)=k_1 $$ have a solution? What are the solutions?

\item % (d)

 Show that if $x_0\ne0$ and $k_0,k_1$  are

arbitrary constants, the initial value problem $$ x^2y''-4xy'+6y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \eqno{\rm (B)} $$ has infinitely many solutions on $(-\infty,\infty)$. On what interval does (B) have a unique solution? \end{alist}

\item\label{exer:5.1.43} %\exercisec \begin{alist} \item % (a) Verify that $y_1=x$ and $y_2=x^2$ satisfy $$ x^2y''-2xy'+2y=0 \eqno{\rm (A)} $$ on $(-\infty,\infty)$ and that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(-\infty,0)$ and $(0,\infty)$. \item % (b) Let $a_1$, $a_2$, $b_1$, and $b_2$ be constants. Show that $$ y=\left\{\begin{array}{rr} a_1x+a_2x^2,&x\ge 0,
b_1x+b_2x^2,&x<0\phantom{,} \end{array}\right. $$ 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.

\item % (c)

 For what values of $k_0$ and  $k_1$

does the initial value problem $$ x^2y''-2xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1 $$ have a solution? What are the solutions?

\item % (d)

 Show that if $x_0\ne0$ and $k_0,k_1$  are

arbitrary constants then the initial value problem $$ x^2y''-2xy'+2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 $$ has a unique solution on $(-\infty,\infty)$. \end{alist}

\item\label{exer:5.1.44} \begin{alist} \item % (a) Verify that $y_1=x^3$ and $y_2=x^4$ satisfy $$ x^2y''-6xy'+12y=0 \eqno{\rm (A)} $$ on $(-\infty,\infty)$, and that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(-\infty,0)$ and $(0,\infty)$. \item % (b) Show that $y$ is a solution of (A) on $(-\infty,\infty)$ if and only if $$ y=\left\{\begin{array}{rr} a_1x^3+a_2x^4,&x\ge 0,
b_1x^3+b_2x^4,&x<0, \end{array}\right. $$ where $a_1$, $a_2$, $b_1$, and $b_2$ are arbitrary constants.

\item % (c)

 For what values of $k_0$ and  $k_1$

does the initial value problem $$ x^2y''-6xy'+12y=0, \quad y(0)=k_0,\quad y'(0)=k_1 $$ have a solution? What are the solutions?

\item % (d)

 Show that if $x_0\ne0$ and $k_0,k_1$  are

arbitrary constants then the initial value problem $$ x^2y''-6xy'+12y=0, \quad y(x_0)=k_0,\quad y'(x_0)=k_1 \eqno{\rm (B)} $$ has infinitely many solutions on $(-\infty,\infty)$. On what interval does (B) have a unique solution? \end{alist}

\end{exerciselist} $$}


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