9.2E: Higher Order Constant Coefficient Homogeneous Equations (Exercises)

Q9.2.1

In Exercises 9.2.1-9.2.14 find the general solution.

1. $y'''-3y''+3y'-y=0$

2. $y^{(4)}+8y''-9y=0$

3. $y'''-y''+16y'-16y=0$

4. $2y'''+3y''-2y'-3y=0$

5. $y'''+5y''+9y'+5y=0$

6. $4y'''-8y''+5y'-y=0$

7. $27y'''+27y''+9y'+y=0$

8. $y^{(4)}+y''=0$

9. $y^{(4)}-16y=0$

10. $y^{(4)}+12y''+36y=0$

11. $16y^{(4)}-72y''+81y=0$

12. $6y^{(4)}+5y'''+7y''+5y'+y=0$

13. $4y^{(4)}+12y'''+3y''-13y'-6y=0$

14. $y^{(4)}-4y'''+7y''-6y'+2y=0$

Q9.2.2

In Exercises 9.2.15-9.2.27 solve the initial value problem. Graph the solution for Exercises 9.2.17-9.2.19 and 9.2.27.

15. $y'''-2y''+4y'-8y=0, \quad y(0)=2,\quad y'(0)=-2,\; y''(0)=0$

16. $y'''+3y''-y'-3y=0, \quad y(0)=0,\quad y'(0)=14,\quad y''(0)=-40$

17. $y'''-y''-y'+y=0, \quad y(0)=-2,\quad y'(0)=9,\quad y''(0)=4$

18. $y'''-2y'-4y=0, \quad y(0)=6,\quad y'(0)=3,\quad y''(0)=22$

19. $3y'''-y''-7y'+5y=0, \quad y(0)= \frac{14}{5},\quad y'(0)=0,\quad y''(0)=10$

20. $y'''-6y''+12y'-8y=0, \quad y(0)=1,\quad y'(0)=-1,\quad y''(0)=-4$

21. $2y'''-11y''+12y'+9y=0, \quad y(0)=6,\quad y'(0)=3,\quad y''(0)=13$

22. $8y'''-4y''-2y'+y=0, \quad y(0)=4,\quad y'(0)=-3,\quad y''(0)=-1$

23. $y^{(4)}-16y=0, \quad y(0)=2,\; y'(0)=2,\; y''(0)=-2,\; y'''(0)=0$

24. $y^{(4)}-6y'''+7y''+6y'-8y=0, \quad y(0)=-2,\quad y'(0)=-8,\quad y''(0)=-14,\quad y'''(0)=-62$

25. $4y^{(4)}-13y''+9y=0, \quad y(0)=1,\quad y'(0)=3,\quad y''(0)=1,\quad y'''(0)=3$

26. $y^{(4)}+2y'''-2y''-8y'-8y=0, \quad y(0)=5,\quad y'(0)=-2,\quad y''(0)=6,\quad y'''(0)=8$

27. $4y^{(4)}+8y'''+19y''+32y'+12y=0, \quad y(0)=3,\quad y'(0)=-3,\quad y''(0)= -\frac{7}{2},\quad y'''(0)=\frac{31}{4}$

Q9.2.3

28. Find a fundamental set of solutions of the given equation, and verify that it is a fundamental set by evaluating its Wronskian at $x=0$.

1. $(D-1)^2(D-2)y=0$
2. $(D^2+4)(D-3)y=0$
3. $(D^2+2D+2)(D-1)y=0$
4. $D^3(D-1)y=0$
5. $(D^2-1)(D^2+1)y=0$
6. $(D^2-2D+2)(D^2+1)y=0$

Q9.2.4

In Exercises 9.2.29-9.2.38 find a fundamental set of solutions.

29. $(D^2+6D+13)(D-2)^2D^3y=0$

30. $(D-1)^2(2D-1)^3(D^2+1)y=0$

31. $(D^2+9)^3D^2y=0$

32. $(D-2)^3(D+1)^2Dy=0$

33. $(D^2+1)(D^2+9)^2(D-2)y=0$

34. $(D^4-16)^2y=0$

35. $(4D^2+4D+9)^3y=0$

36. $D^3(D-2)^2(D^2+4)^2y=0$

37. $(4D^2+1)^2(9D^2+4)^3y=0$

38. $\left[(D-1)^4-16\right]y=0$

Q9.2.5

39 It can be shown that $$\left|\begin{array}{cccc} 1&1&\cdots&1\\[4pt] a_1&a_2&\cdots&a_n\\[4pt] a^2_1&a^2_2&\cdots&a^2_n\\[4pt] \vdots&\vdots&\ddots&\vdots\\[4pt] a^{n-1}_1&a^{n-1}_2&\cdots&a^{n-1}_n\end{array}\right|= \prod_{1\leq i<j\leq n}(a_{j}-a_{i}),\tag{A}$$

where the left side is the Vandermonde determinant and the right side is the product of all factors of the form $(a_j-a_i)$ with $i$ and $j$ between $1$ and $n$ and $i<j$.>

1. Verify (A) for $n=2$ and $n=3$.
2. Find the Wronskian of $\{e^{{a_1}x}, \quad e^{{a_2}x},\dots, e^{{a_n}x}\}$.

40. A theorem from algebra says that if $P_1$ and $P_2$ are polynomials with no common factors then there are polynomials $Q_1$ and $Q_2$ such that $$Q_1P_1+Q_2P_2=1.\nonumber$$ This implies that $$Q_1(D)P_1(D)y+Q_2(D)P_2(D)y=y\nonumber$$ for every function $y$ with enough derivatives for the left side to be defined.

1. Use this to show that if $P_1$ and $P_2$ have no common factors and $$P_1(D)y=P_2(D)y=0\nonumber$$ then $y=0$.
2. Suppose $P_1$ and $P_2$ are polynomials with no common factors. Let $u_1$, …, $u_r$ be linearly independent solutions of $P_1(D)y=0$ and let $v_1$, …, $v_s$ be linearly independent solutions of $P_2(D)y=0$. Use (a) to show that $\{u_1,\dots,u_r,\: v_1,\dots,v_s\}$ is a linearly independent set.
3. Suppose the characteristic polynomial of the constant coefficient equation $$a_0y^{(n)}+a_1y^{(n-1)}+\cdots+a_ny=0 \tag{A}$$ has the factorization $$p(r)=a_0p_1(r)p_2(r)\cdots p_k(r),\nonumber$$ where each $p_j$ is of the form $$p_j(r)=(r-r_j)^{n_j} \mbox{ or } p_j(r)=[(r-\lambda_j)^2+w^2_j]^{m_j}\quad (\omega_j>0)\nonumber$$ and no two of the polynomials $p_1$, $p_2$, …, $p_k$ have a common factor. Show that we can find a fundamental set of solutions $\{y_1,y_2,\dots,y_n\}$ of (A) by finding a fundamental set of solutions of each of the equations $$p_j(D)y=0,\quad 1\le j\le k,\nonumber$$ and taking $\{y_1,y_2,\dots,y_n\}$ to be the set of all functions in these separate fundamental sets.

41.

1. Show that if $$z=p(x)\cos\omega x+q(x)\sin\omega x, \tag{A}$$ where $p$ and $q$ are polynomials of degree $\le k$, then $$(D^2+\omega^2)z=p_1(x)\cos\omega x+q_1(x)\sin\omega x,\nonumber$$ where $p_1$ and $q_1$ are polynomials of degree $\le k-1$.
2. Apply (a) $m$ times to show that if $z$ is of the form (A) where $p$ and $q$ are polynomial of degree $\le m-1$, then $$(D^2+\omega^2)^mz=0. \tag{B}$$
3. Use Equation 9.2.17 to show that if $y=e^{\lambda x}z$ then $$[(D-\lambda)^2+\omega^2]^my=e^{\lambda x}(D^2+\omega^2)^mz.\nonumber$$
4. Conclude from (b) and (c) that if $p$ and $q$ are arbitrary polynomials of degree $\le m-1$ then $$y=e^{\lambda x}(p(x)\cos\omega x+q(x)\sin\omega x)\nonumber$$ is a solution of $$[(D-\lambda)^2+\omega^2]^my=0. \tag{C}$$
5. Conclude from (d) that the functions $$\begin{array}{rl} e^{\lambda x}\cos\omega x, xe^{\lambda x}\cos\omega x, &\dots, x^{m-1}e^{\lambda x}\cos\omega x,\\[4pt] e^{\lambda x}\sin\omega x, xe^{\lambda x}\sin\omega x,& \dots, x^{m-1}e^{\lambda x}\sin\omega x \end{array} \tag{D}$$ are all solutions of (C).
6. Complete the proof of Theorem 9.2.2 by showing that the functions in (D) are linearly independent.

42.

1. Use the trigonometric identities $$\begin{aligned} \cos(A+B)&=\cos A\cos B-\sin A\sin B\\[4pt] \sin(A+B)&=\cos A\sin B+\sin A\cos B\end{aligned}\nonumber$$ to show that $$(\cos A+i\sin A)(\cos B+i\sin B)=\cos(A+B)+i\sin(A+B).\nonumber$$
2. Apply (a) repeatedly to show that if $n$ is a positive integer then $$\prod_{k=1}^n(\cos A_k+i\sin A_k)=\cos(A_1+A_2+\cdots+A_n) +i\sin(A_1+A_2+\cdots+A_n).\nonumber$$
3. Infer from (b) that if $n$ is a positive integer then $$(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta. \tag{A}$$
4. Show that (A) also holds if $n=0$ or a negative integer. HINT: Verify by direct calculation that $$(\cos\theta +i\sin\theta )^{-1}=(\cos\theta -i\sin\theta ).\nonumber$$ Then replace $\theta$ by $-\theta$ in $(A)$.
5. Now suppose $n$ is a positive integer. Infer from (A) that if $$z_k=\cos\left(2k\pi\over n\right)+i\sin\left(2k\pi\over n\right) ,\quad k=0,1,\dots,n-1,\nonumber$$ and $$\zeta_k=\cos\left((2k+1)\pi\over n\right)+i\sin\left((2k+1)\pi\over n\right) ,\quad k=0,1,\dots,n-1,\nonumber$$ then $$z_k^n=1\quad\mbox{ and }\quad\zeta_k^n=-1,\quad k=0,1,\dots,n-1.\nonumber$$ (Why don’t we also consider other integer values for $k$?)
6. Let $\rho$ be a positive number. Use (e) to show that $$z^n-\rho=(z-\rho^{1/n} z_0)(z-\rho^{1/n}z_1)\cdots(z-\rho^{1/n} z_{n-1})\nonumber$$ and $$z^n+\rho=(z-\rho^{1/n} \zeta_0)(z-\rho^{1/n} \zeta_1)\cdots(z-\rho^{1/n} \zeta_{n-1}).\nonumber$$

43. Use (e) of Exercise 9.2.42 to find a fundamental set of solutions of the given equation.

1. $y'''-y=0$
2. $y'''+y=0$
3. $y^{(4)}+64y=0$
4. $y^{(6)}-y=0$
5. $y^{(6)}+64y=0$
6. $\left[(D-1)^6-1\right]y=0$
7. $y^{(5)}+y^{(4)}+y'''+y''+y'+y=0$

44. An equation of the form $$a_0x^ny^{(n)}+a_1x^{n-1}y^{(n-1)}+\cdots +a_{n-1}xy'+a_ny=0,\quad x>0, \tag{A}$$ where $a_0$, $a_1$, …, $a_n$ are constants, is an Euler or equidimensional equation. Show that if $$x=e^t \quad \mbox{ and } \quad Y(t)=y(x(t)), \tag{B}$$ then

$$\begin{aligned} x\frac{dy}{dx}&=\frac{dY}{dt} \\[4pt] x^{2}\frac{d^{2}y}{dx^{2}}&=\frac{d^{2}Y}{dt^{2}}-\frac{dY}{dt} \\[4pt] x^{3}\frac{d^{3}y}{dx^{3}}&=\frac{d^{3}Y}{dt^{3}}-3\frac{d^{2}Y}{dt^{2}}+2\frac{dY}{dt}. \end{aligned} \nonumber$$

In general, it can be shown that if $r$ is any integer $\ge2$ then

$$x^r {d^ry\over dx^r}={d^rY\over dt^r}+ A_{1r}{d^{r-1}Y\over dt^{r-1}}+\cdots+A_{r-1,r} {dY\over dt}\nonumber$$

where $A_{1r}$, …, $A_{r-1,r}$ are integers. Use these results to show that the substitution (B) transforms (A) into a constant coefficient equation for $Y$ as a function of $t$.

45. Use Exercise 9.2.44 to show that a function $y=y(x)$ satisfies the equation $$a_0x^3y'''+a_1x^2y''+a_2xy'+a_3y=0, \tag{A}$$ on $(0,\infty)$ if and only if the function $Y(t)=y(e^t)$ satisfies $$a_0{d^3Y\over dt^3}+(a_1-3a_0) {d^2Y\over dt^2}+(a_2-a_1+2a_0) {dY\over dt}+a_3Y=0.\nonumber$$ Assuming that $a_0$, $a_1$, $a_2$, $a_3$ are real and $a_0 \ne0$, find the possible forms for the general solution of (A).