5.1E: Homogeneous Linear Equations (Exercises)

Q5.1.1

1.

1. Verify that $y_1=e^{2x}$ and $y_2=e^{5x}$ are solutions of $$y''-7y'+10y=0 \tag{A}$$ on $(-\infty,\infty)$.
2. 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 $(-\infty,\infty)$.
3. Solve the initial value problem $$y''-7y'+10y=0,\quad y(0)=-1,\quad y'(0)=1.\nonumber$$
4. Solve the initial value problem $$y''-7y'+10y=0,\quad y(0)=k_0,\quad y'(0)=k_1.\nonumber$$

2.

1. Verify that $y_1=e^x\cos x$ and $y_2=e^x\sin x$ are solutions of $$y''-2y'+2y=0 \tag{A}$$ on $(-\infty,\infty)$.
2. 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 $(-\infty,\infty)$.
3. Solve the initial value problem $$y''-2y'+2y=0,\quad y(0)=3,\quad y'(0)=-2.\nonumber$$
4. Solve the initial value problem $$y''-2y'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1.\nonumber$$

3.

1. Verify that $y_1=e^x$ and $y_2=xe^x$ are solutions of $$y''-2y'+y=0 \tag{A}$$ on $(-\infty,\infty)$.
2. 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 $(-\infty,\infty)$.
3. Solve the initial value problem $$y''-2y'+y=0,\quad y(0)=7,\quad y'(0)=4.\nonumber$$
4. Solve the initial value problem $$y''-2y'+y=0,\quad y(0)=k_0,\quad y'(0)=k_1.\nonumber$$

4.

1. Verify that $y_1=1/(x-1)$ and $y_2=1/(x+1)$ are solutions of $$(x^2-1)y''+4xy'+2y=0 \tag{A}$$ on $(-\infty,-1)$, $(-1,1)$, and $(1,\infty)$. What is the general solution of (A) on each of these intervals?
2. 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?
3. Graph the solution of the initial value problem.
4. Verify Abel’s formula for $y_1$ and $y_2$, with $x_0=0$.

5. Compute the Wronskians of the given sets of functions.

1. $\{1, e^{x}\}$
2. $\{e^{x}, e^{x}\sin x\}$
3. $\{x+1, x^{2}+2\}$
4. $\{x^{1/2}, x^{-1/3}\}$
5. $\{\frac{\sin x}{x},\frac{\cos x}{x}\}$
6. $\{x\ln |x|, x^{2}\ln |x|\}$
7. $\{e^{x}\cos\sqrt{x}, e^{x}\sin\sqrt{x}\}$

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 $W(\pi)=0$.

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.)

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.)

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

$$y''+p(x)y'+q(x)y=0 \tag{A}$$

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

1. Show that if $K$ is an arbitrary nonzero constant and $y_2$ satisfies $$y_1y_2'-y_1'y_2=Ke^{-P(x)} \tag{B}$$ 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 $(a,b)$.
2. 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 $(a,b)$.

Q5.1.2

In Exercises 5.1.10-5.1.23 use the method suggested by Exercise 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 $y_{2}$.

10. $y''-2y'-3y=0$; $y_1=e^{3x}$

11. $y''-6y'+9y=0$; $y_1=e^{3x}$

12. $y''-2ay'+a^2y=0$ ($a=$ constant); $y_1=e^{ax}$

13. $x^2y''+xy'-y=0$; $y_1=x$

14. $x^2y''-xy'+y=0$; $y_1=x$

15. $x^2y''-(2a-1)xy'+a^2y=0$ ($a=$ nonzero constant);  $x>0$; $y_1=x^a$

16. $4x^2y''-4xy'+(3-16x^2)y=0$; $y_1=x^{1/2}e^{2x}$

17. $(x-1)y''-xy'+y=0$; $y_1=e^x$

18. $x^2y''-2xy'+(x^2+2)y=0$; $y_1=x\cos x$

19. $4x^2(\sin x)y''-4x(x\cos x+\sin x)y'+(2x\cos x+3\sin x)y=0$; $y_1=x^{1/2}$

20. $(3x-1)y''-(3x+2)y'-(6x-8)y=0$; $y_1=e^{2x}$

21. $(x^2-4)y''+4xy'+2y=0$; $y_1={1\over x-2}$

22. $(2x+1)xy''-2(2x^2-1)y'-4(x+1)y=0$;$y_1={1\over x}$

23. $(x^2-2x)y''+(2-x^2)y'+(2x-2)y=0$;$y_1=e^x$

Q5.1.3

24. Suppose $p$ and $q$ are continuous on an open interval $(a,b)$ and let $x_0$ be in $(a,b)$. Use Theorem 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 $y\equiv0$.

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 $(a,b)$.

1. 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 $(a,b)$.
2. 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 $(a,b)$.

26. Suppose $p$ and $q$ are continuous on $(a,b)$ and $y_1$ and $y_2$ are solutions of

$$y''+p(x)y'+q(x)y=0 \tag{A}$$

on $(a,b)$. Let

$$z_1=\alpha y_1+\beta y_2\quad\text{ and} \quad z_2=\gamma y_1+\delta y_2,\nonumber$$

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 $\{y_1,y_2\}$.

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

$$y''+p(x)y'+q(x)y=0 \tag{A}$$

on $(a,b)$. Let

$$z_1=\alpha y_1+\beta y_2\quad\text{ and} \quad z_2=\gamma y_1+\delta y_2,\nonumber$$

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 $\alpha\gamma-\beta\delta\ne0$.

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 $(a,b)$.

29. Let

$$y_1=x^3\quad\mbox{ and }\quad y_2=\left\{\begin{array}{rl} x^3,&x\ge 0,\\[4pt] -x^3,&x<0.\end{array}\right.\nonumber$$

1. Show that the Wronskian of $\{y_1,y_2\}$ is defined and identically zero on $(-\infty,\infty)$.
2. Suppose $a<0<b$. Show that $\{y_1,y_2\}$ is linearly independent on $(a,b)$.
3. Use Exercise 5.1.25b to show that these results don’t contradict Theorem 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 $(a,b)$.

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 $(a,b)$.

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 $(x_1,x_2)$.

32. Suppose $p$ and $q$ are continuous on $(a,b)$ and every solution of

$$y''+p(x)y'+q(x)y=0 \tag{A}$$

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

33. Suppose $p_1$, $p_2$, $q_1$, and $q_2$ are continuous on $(a,b)$ and the equations

$$y''+p_1(x)y'+q_1(x)y=0 \quad \text{and} \quad y''+p_2(x)y'+q_2(x)y=0\nonumber$$

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

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

$$\frac{1}{W} \left| \begin{array}{ccc} y & y_1 & y_2 \\[4pt] y' & y'_1 & y'_2 \\[4pt] y'' & y_1'' & y_2'' \end{array} \right|=0\nonumber$$

can be written as

$$y''+p(x)y'+q(x)y=0, \tag{A}$$

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

35. Use the method suggested by Exercise 5.1.34 to find a linear homogeneous equation for which the given functions form a fundamental set of solutions on some interval.

1. $e^{x}\cos 2x, e^{x}\sin 2x$
2. $x, e^{2x}$
3. $x, x\ln x$
4. $\cos (\ln x), \sin (\ln x)$
5. $\cosh x, \sinh x$
6. $x^{2}-1, x^{2}+1$

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

$$y''+p(x)y'+q(x)y=0 \tag{A}$$

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 $(a,b)$.

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

$$y''+p(x)y'+q(x)y=0 \tag{A}$$

such that

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

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

1. Show that $\{y_1,y_2\}$ is linearly independent on $(a,b)$.
2. Show that an arbitrary solution $y$ of (A) on $(a,b)$ can be written as $y=y(x_0)y_1+y'(x_0)y_2$.
3. 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 $y_2$.

38. Find solutions $y_1$ and $y_2$ of the equation $y''=0$ that satisfy the initial conditions

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

Then use Exercise 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 $y_2$.

39. Let $x_0$ be an arbitrary real number. Given (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

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

Then use Exercise 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 $y_2$.

40. Let $x_0$ be an arbitrary real number. Given (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

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

Then use Exercise 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

$$\begin{aligned} \cos(A+B)&=\cos A\cos B-\sin A\sin B\\[4pt] \sin(A+B)&=\sin A\cos B+\cos A\sin B\end{aligned}\nonumber$$

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

41. Recall from Exercise 5.1.4 that $1/(x-1)$ and $1/(x+1)$ are solutions of

$$(x^2-1)y''+4xy'+2y=0 \tag{A}$$

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

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

Then use Exercise 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 $y_2$.

42.

1. Verify that $y_1=x^2$ and $y_2=x^3$ satisfy $$x^2y''-4xy'+6y=0 \tag{A}$$ on $(-\infty,\infty)$ and that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(-\infty,0)$ and $(0,\infty)$.
2. Let $a_1$, $a_2$, $b_1$, and $b_2$ be constants. Show that $$y=\left\{\begin{array}{rr} a_1x^2+a_2x^3,&x\ge 0,\\[4pt] b_1x^2+b_2x^3,&x<0\phantom{,} \end{array}\right.\nonumber$$ 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 $$y=\left\{\begin{array}{rr} c_1x^2+c_2x^3,&x\ge 0,\\[4pt] c_1x^2+c_3x^3,&x<0, \end{array}\right.\nonumber$$ where $c_1$, $c_2$, and $c_3$ are arbitrary constants.
3. 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?
4. Show that if $x_0\ne0$ and $k_0,k_1$ are arbitrary constants, the initial value problem $$x^2y''-4xy'+6y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \tag{B}$$ has infinitely many solutions on $(-\infty,\infty)$. On what interval does (B) have a unique solution?

43.

1. Verify that $y_1=x$ and $y_2=x^2$ satisfy $$x^2y''-2xy'+2y=0 \tag{A}$$ on $(-\infty,\infty)$ and that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(-\infty,0)$ and $(0,\infty)$.
2. Let $a_1$, $a_2$, $b_1$, and $b_2$ be constants. Show that $$y=\left\{\begin{array}{rr} a_1x+a_2x^2,&x\ge 0,\\[4pt] b_1x+b_2x^2,&x<0\phantom{,} \end{array}\right.\nonumber$$ 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.
3. 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?
4. 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 $(-\infty,\infty)$.

44.

1. Verify that $y_1=x^3$ and $y_2=x^4$ satisfy $$x^2y''-6xy'+12y=0 \tag{A}$$ on $(-\infty,\infty)$, and that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on $(-\infty,0)$ and $(0,\infty)$.
2. Show that $y$ is a solution of (A) on $(-\infty,\infty)$ if and only if $$y=\left\{\begin{array}{rr} a_1x^3+a_2x^4,&x\ge 0,\\[4pt] b_1x^3+b_2x^4,&x<0, \end{array}\right.\nonumber$$ where $a_1$, $a_2$, $b_1$, and $b_2$ are arbitrary constants.
3. 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?
4. Show that if $x_0\ne0$ and $k_0,k_1$ are arbitrary constants then the initial value problem $$x^2y''-6xy'+12y=0, \quad y(x_0)=k_0,\quad y'(x_0)=k_1 \tag{B}$$ has infinitely many solutions on $(-\infty,\infty)$. On what interval does (B) have a unique solution?