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# 5.1E: Homogeneous Linear Equations (Exercises)

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## 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

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

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

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

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\\ \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 $y=\left\{\begin{array}{rr} a_1x^2+a_2x^3,&x\ge 0,\\ b_1x^2+b_2x^3,&x<0\phantom{,} \end{array}\right.\nonumber$  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.
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 $y=\left\{\begin{array}{rr} a_1x+a_2x^2,&x\ge 0,\\ b_1x+b_2x^2,&x<0\phantom{,} \end{array}\right.\nonumber$  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 $y=\left\{\begin{array}{rr} a_1x^3+a_2x^4,&x\ge 0,\\ b_1x^3+b_2x^4,&x<0, \end{array}\right.\nonumber$  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?