Processing math: 68%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

5.1E: Homogeneous Linear Equations (Exercises)

( \newcommand{\kernel}{\mathrm{null}\,}\)

Q5.1.1

1.

  1. Verify that y1=e2x and y2=e5x are solutions of y7y+10y=0 on (,).
  2. Verify that if c1 and c2 are arbitrary constants then y=c1e2x+c2e5x is a solution of (A) on (,).
  3. Solve the initial value problem y7y+10y=0,y(0)=1,y(0)=1.
  4. Solve the initial value problem y7y+10y=0,y(0)=k0,y(0)=k1.

2.

  1. Verify that y1=excosx and y2=exsinx are solutions of y2y+2y=0 on (,).
  2. Verify that if c1 and c2 are arbitrary constants then y=c1excosx+c2exsinx is a solution of (A) on (,).
  3. Solve the initial value problem y2y+2y=0,y(0)=3,y(0)=2.
  4. Solve the initial value problem y2y+2y=0,y(0)=k0,y(0)=k1.

3.

  1. Verify that y1=ex and y2=xex are solutions of y2y+y=0 on (,).
  2. Verify that if c1 and c2 are arbitrary constants then y=ex(c1+c2x) is a solution of (A) on (,).
  3. Solve the initial value problem y2y+y=0,y(0)=7,y(0)=4.
  4. Solve the initial value problem y2y+y=0,y(0)=k0,y(0)=k1.

4.

  1. Verify that y1=1/(x1) and y2=1/(x+1) are solutions of (x21)y+4xy+2y=0 on (,1), (1,1), and (1,). What is the general solution of (A) on each of these intervals?
  2. Solve the initial value problem (x21)y+4xy+2y=0,y(0)=5,y(0)=1. What is the interval of validity of the solution?
  3. Graph the solution of the initial value problem.
  4. Verify Abel’s formula for y1 and y2, with x0=0.

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

  1. {1,ex}
  2. {ex,exsinx}
  3. {x+1,x2+2}
  4. {x1/2,x1/3}
  5. {sinxx,cosxx}
  6. {xln|x|,x2ln|x|}
  7. {excosx,exsinx}

6. Find the Wronskian of a given set {y1,y2} of solutions of

y+3(x2+1)y2y=0,

given that W(π)=0.

7. Find the Wronskian of a given set {y1,y2} of solutions of

(1x2)y2xy+α(α+1)y=0,

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

8. Find the Wronskian of a given set {y1,y2} of solutions of

x2y+xy+(x2ν2)y=0,

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 y1 is a solution of

y+p(x)y+q(x)y=0

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

  1. Show that if K is an arbitrary nonzero constant and y2 satisfies y1y2y1y2=KeP(x) on (a,b), then y2 also satisfies (A) on (a,b), and {y1,y2} is a fundamental set of solutions on (A) on (a,b).
  2. Conclude from (a) that if y2=uy1 where u=KeP(x)y21(x), then {y1,y2} 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 y2 that isn’t a constant multiple of the solution y1. Choose K conveniently to simplify y2.

10. y2y3y=0; y1=e3x

11. y6y+9y=0; y1=e3x

12. y2ay+a2y=0 (a= constant); y1=eax

13. x2y+xyy=0; y1=x

14. x2yxy+y=0; y1=x

15. x2y(2a1)xy+a2y=0 (a= nonzero constant);  x>0; y1=xa

16. 4x2y4xy+(316x2)y=0; y1=x1/2e2x

17. (x1)yxy+y=0; y1=ex

18. x2y2xy+(x2+2)y=0; y1=xcosx

19. 4x2(sinx)y4x(xcosx+sinx)y+(2xcosx+3sinx)y=0; y1=x1/2

20. (3x1)y(3x+2)y(6x8)y=0; y1=e2x

21. (x24)y+4xy+2y=0; y1=1x2

22. (2x+1)xy2(2x21)y4(x+1)y=0;y1=1x

23. (x22x)y+(2x2)y+(2x2)y=0;y1=ex

Q5.1.3

24. Suppose p and q are continuous on an open interval (a,b) and let x0 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,y(x0)=0,y(x0)=0

on (a,b) is the trivial solution y0.

25. Suppose P0, P1, and P2 are continuous on (a,b) and let x0 be in (a,b). Show that if either of the following statements is true then P0(x)=0 for some x in (a,b).

  1. The initial value problem P0(x)y+P1(x)y+P2(x)y=0,y(x0)=k0,y(x0)=k1 has more than one solution on (a,b).
  2. The initial value problem P0(x)y+P1(x)y+P2(x)y=0,y(x0)=0,y(x0)=0 has a nontrivial solution on (a,b).

26. Suppose p and q are continuous on (a,b) and y1 and y2 are solutions of

y+p(x)y+q(x)y=0

on (a,b). Let

z1=αy1+βy2 andz2=γy1+δy2,

where α, β, γ, and δ are constants. Show that if {z1,z2} is a fundamental set of solutions of (A) on (a,b) then so is {y1,y2}.

27. Suppose p and q are continuous on (a,b) and {y1,y2} is a fundamental set of solutions of

y+p(x)y+q(x)y=0

on (a,b). Let

z1=αy1+βy2 andz2=γy1+δy2,

where α,β,γ, and δ are constants. Show that {z1,z2} is a fundamental set of solutions of (A) on (a,b) if and only if αγβδ0.

28. Suppose y1 is differentiable on an interval (a,b) and y2=ky1, where k is a constant. Show that the Wronskian of {y1,y2} is identically zero on (a,b).

29. Let

y1=x3 and y2={x3,x0,x3,x<0.

  1. Show that the Wronskian of {y1,y2} is defined and identically zero on (,).
  2. Suppose a<0<b. Show that {y1,y2} 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 y1 nor y2 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).

30. Suppose p and q are continuous on (a,b) and {y1,y2} is a set of solutions of

y+p(x)y+q(x)y=0

on (a,b) such that either y1(x0)=y2(x0)=0 or y1(x0)=y2(x0)=0 for some x0 in (a,b). Show that {y1,y2} is linearly dependent on (a,b).

31. Suppose p and q are continuous on (a,b) and {y1,y2} is a fundamental set of solutions of

y+p(x)y+q(x)y=0

on (a,b). Show that if y1(x1)=y1(x2)=0, where a<x1<x2<b, then y2(x)=0 for some x in (x1,x2).

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

y+p(x)y+q(x)y=0

on (a,b) can be written as a linear combination of the twice differentiable functions {y1,y2}. Use Theorem 5.1.1 to show that y1 and y2 are themselves solutions of (A) on (a,b).

33. Suppose p1, p2, q1, and q2 are continuous on (a,b) and the equations

y+p1(x)y+q1(x)y=0andy+p2(x)y+q2(x)y=0

have the same solutions on (a,b). Show that p1=p2 and q1=q2 on (a,b).

34. (For this exercise you have to know about 3×3 determinants.) Show that if y1 and y2 are twice continuously differentiable on (a,b) and the Wronskian W of {y1,y2} has no zeros in (a,b) then the equation

1W|yy1y2yy1y2yy1y2|=0

can be written as

y+p(x)y+q(x)y=0,

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?

This page titled 5.1E: Homogeneous Linear Equations (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

Support Center

How can we help?