5.4E: The Method of Undetermined Coefficients I (Exercises)

Q5.4.1

In Exercises 5.4.1-5.4.14 find a particular solution.

1. \(y''-3y'+2y=e^{3x}(1+x)\)

2. \(y''-6y'+5y=e^{-3x}(35-8x)\)

3. \(y''-2y'-3y=e^x(-8+3x)\)

4. \(y''+2y'+y=e^{2x}(-7-15x+9x^2)\)

5. \(y''+4y=e^{-x}(7-4x+5x^2)\)

6. \(y''-y'-2y=e^x(9+2x-4x^2)\)

7. \(y''-4y'-5y=-6xe^{-x}\)

8. \(y''-3y'+2y=e^x(3-4x)\)

9. \(y''+y'-12y=e^{3x}(-6+7x)\)

10. \(2y''-3y'-2y=e^{2x}(-6+10x)\)

11. \(y''+2y'+y=e^{-x}(2+3x)\)

12. \(y''-2y'+y=e^x(1-6x)\)

13. \(y''-4y'+4y=e^{2x}(1-3x+6x^2)\)

14. \(9y''+6y'+y=e^{-x/3}(2-4x+4x^2)\)

Q5.4.2

In Exercises 5.4.15-5.4.19 find the general solution.

15. \(y''-3y'+2y=e^{3x}(1+x)\)

16. \(y''-6y'+8y=e^x(11-6x)\)

17. \(y''+6y'+9y=e^{2x}(3-5x)\)

18. \(y''+2y'-3y=-16xe^x\)

19. \(y''-2y'+y=e^x(2-12x)\)

Q5.4.3

In Exercises 5.4.20-5.4.23 solve the initial value problem and plot the solution.

20. \(y''-4y'-5y=9e^{2x}(1+x), \quad y(0)=0,\quad y'(0)=-10\)

21. \(y''+3y'-4y=e^{2x}(7+6x), \quad y(0)=2,\quad y'(0)=8\)

22. \(y''+4y'+3y=-e^{-x}(2+8x), \quad y(0)=1,\quad y'(0)=2\)

23. \(y''-3y'-10y=7e^{-2x}, \quad y(0)=1,\quad y'(0)=-17\)

Q5.4.4

In Exercises 5.4.24-5.4.29 use the principle of superposition to find a particular solution.

24. \(y''+y'+y=xe^x+e^{-x}(1+2x)\)

25. \(y''-7y'+12y=-e^x(17-42x)-e^{3x}\)

26. \(y''-8y'+16y=6xe^{4x}+2+16x+16x^2\)

27. \(y''-3y'+2y=-e^{2x}(3+4x)-e^x\)

28. \(y''-2y'+2y=e^x(1+x)+e^{-x}(2-8x+5x^2)\)

29. \(y''+y=e^{-x}(2-4x+2x^2)+e^{3x}(8-12x-10x^2)\)

Q5.4.5

30.

1. Prove that \(y\) is a solution of the constant coefficient equation \[ay''+by'+cy=e^{\alpha x}G(x) \tag{A} \] if and only if \(y=ue^{\alpha x}\), where \(u\) satisfies \[au''+p'(\alpha)u'+p(\alpha)u=G(x) \tag{B} \] and \(p(r)=ar^2+br+c\) is the characteristic polynomial of the complementary equation \[ay''+by'+cy=0.\nonumber \] For the rest of this exercise, let \(G\) be a polynomial. Give the requested proofs for the case where \[G(x)=g_0+g_1x+g_2x^2+g_3x^3.\nonumber \]
2. Prove that if \(e^{\alpha x}\) isn’t a solution of the complementary equation then (B) has a particular solution of the form \(u_p=A(x)\), where \(A\) is a polynomial of the same degree as \(G\), as in Example 5.4.4. Conclude that (A) has a particular solution of the form \(y_p=e^{\alpha x}A(x)\).
3. Show that if \(e^{\alpha x}\) is a solution of the complementary equation and \(xe^{\alpha x}\) isn’t , then (B) has a particular solution of the form \(u_p=xA(x)\), where \(A\) is a polynomial of the same degree as \(G\), as in Example 5.4.5. Conclude that (A) has a particular solution of the form \(y_p=xe^{\alpha x}A(x)\).
4. Show that if \(e^{\alpha x}\) and \(xe^{\alpha x}\) are both solutions of the complementary equation then (B) has a particular solution of the form \(u_p=x^2A(x)\), where \(A\) is a polynomial of the same degree as \(G\), and \(x^2A(x)\) can be obtained by integrating \(G/a\) twice, taking the constants of integration to be zero, as in Example 5.4.6. Conclude that (A) has a particular solution of the form \(y_p=x^2e^{\alpha x}A(x)\).

Q5.4.6

Exercises 5.4.31–5.4.36 treat the equations considered in Examples 5.4.1–5.4.6. Substitute the suggested form of \(y_{p}\) into the equation and equate the resulting coefficients of like functions on the two sides of the resulting equation to derive a set of simultaneous equations for the coefficients in \(y_{p}\). Then solve for the coefficients to obtain \(y_{p}\). Compare the work you’ve done with the work required to obtain the same results in Examples 5.4.1–5.4.6.

31. Compare with Example 5.4.1:

\[y''-7y'+12y=4e^{2x};\quad y_p=Ae^{2x}\nonumber \]

32. Compare with Example 5.4.2:

\[y''-7y'+12y=5e^{4x};\quad y_p=Axe^{4x}\nonumber \]

33. Compare with Example 5.4.3:

\[y''-8y'+16y=2e^{4x};\quad y_p=Ax^2e^{4x}\nonumber \]

34. Compare with Example 5.4.4:

\[y''-3y'+2y=e^{3x}(-1+2x+x^2),\quad y_p=e^{3x}(A+Bx+Cx^2)\nonumber \]

35. Compare with Example 5.4.5:

\[y''-4y'+3y=e^{3x}(6+8x+12x^2),\quad y_p=e^{3x}(Ax+Bx^2+Cx^3)\nonumber \]

36. Compare with Example 5.4.6:

\[4y''+4y'+y=e^{-x/2}(-8+48x+144x^2),\quad y_p=e^{-x/2}(Ax^2+Bx^3+Cx^4)\nonumber \]

Q5.4.7

37. Write \(y=ue^{\alpha x}\) to find the general solution.

1. \(y''+2y'+y={e^{-x}\over\sqrt x}\)
2. \(y''+6y'+9y=e^{-3x}\ln x\)
3. \(y''-4y'+4y={e^{2x}\over1+x}\)
4. \(4y''+4y'+y={4e^{-x/2}\left({1\over x}+x\right)}\)

38. Suppose \(\alpha\ne0\) and \(k\) is a positive integer. In most calculus books integrals like \(\int x^k e^{\alpha x}\,dx\) are evaluated by integrating by parts \(k\) times. This exercise presents another method. Let

\[y=\int e^{\alpha x}P(x)\,dx\nonumber \]

with

\[P(x)=p_0+p_1x+\cdots+p_kx^k\nonumber \]

(where \(p_k \neq 0\)).

1. Show that \(y=e^{\alpha x}u\), where \[u'+\alpha u=P(x). \tag{A} \]
2. Show that (A) has a particular solution of the form \[u_p=A_0+A_1x+\cdots+A_kx^k,\nonumber \] where \(A_k\), \(A_{k-1}\), …, \(A_0\) can be computed successively by equating coefficients of \(x^k,x^{k-1}, \dots,1\) on both sides of the equation \[u_p'+\alpha u_p=P(x).\nonumber \]
3. Conclude that \[\int e^{\alpha x}P(x)\,dx=\left(A_0+A_1x+\cdots+A_kx^k\right)e^{\alpha x} +c,\nonumber \] where \(c\) is a constant of integration.

39. Use the method of Exercise 5.4.38 to evaluate the integral.

1. \(\int e^{x}(4+x)dx\)
2. \(\int e^{-x}(-1+x^{2})dx\)
3. \(\int x^{3}e^{-2x}dx\)
4. \(\int e^{x}(1+x)^{2}dx\)
5. \(\int e^{3x}(-14+30x+27x^{2})dx\)
6. \(\int e^{-x}(1+6x^{2}-14x^{3}+3x^{4})dx\)

40. Use the method suggested in Exercise 5.4.38 to evaluate \(\int x^ke^{\alpha x}\,dx\), where \(k\) is an arbitrary positive integer and \(\alpha\ne0\).