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

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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In Exercises [exer:9.2.1}– [exer:9.2.14} find the general solution.

[exer:9.2.1] $$y'''-3y''+3y'-y=0$$

[exer:9.2.2] $$y^{(4)}+8y''-9y=0$$

[exer:9.2.3] $$y'''-y''+16y'-16y=0$$

[exer:9.2.4] $$2y'''+3y''-2y'-3y=0$$

[exer:9.2.5] $$y'''+5y''+9y'+5y=0$$

[exer:9.2.6] $$4y'''-8y''+5y'-y=0$$

[exer:9.2.7] $$27y'''+27y''+9y'+y=0$$

[exer:9.2.8] $$y^{(4)}+y''=0$$

[exer:9.2.9] $$y^{(4)}-16y=0$$

[exer:9.2.10] $$y^{(4)}+12y''+36y=0$$

[exer:9.2.11] $$16y^{(4)}-72y''+81y=0$$

[exer:9.2.12] $$6y^{(4)}+5y'''+7y''+5y'+y=0$$

[exer:9.2.13] $$4y^{(4)}+12y'''+3y''-13y'-6y=0$$

[exer:9.2.14] $$y^{(4)}-4y'''+7y''-6y'+2y=0$$

[exer:9.2.15] $$y'''-2y''+4y'-8y=0, \quad y(0)=2,\quad y'(0)=-2,\; y''(0)=0$$

[exer:9.2.16] $$y'''+3y''-y'-3y=0, \quad y(0)=0,\quad y'(0)=14,\quad y''(0)=-40$$

[exer:9.2.17] $$y'''-y''-y'+y=0, \quad y(0)=-2,\quad y'(0)=9,\quad y''(0)=4$$

[exer:9.2.18] $$y'''-2y'-4y=0, \quad y(0)=6,\quad y'(0)=3,\quad y''(0)=22$$

[exer:9.2.19] $$3y'''-y''-7y'+5y=0, \quad y(0)= 14\over5,\quad y'(0)=0,\quad y''(0)=10$$

[exer:9.2.20] $$y'''-6y''+12y'-8y=0, \quad y(0)=1,\quad y'(0)=-1,\quad y''(0)=-4$$

[exer:9.2.21] $$2y'''-11y''+12y'+9y=0, \quad y(0)=6,\quad y'(0)=3,\quad y''(0)=13$$

[exer:9.2.22] $$8y'''-4y''-2y'+y=0, \quad y(0)=4,\quad y'(0)=-3,\quad y''(0)=-1$$

[exer:9.2.23] $$y^{(4)}-16y=0, \quad y(0)=2,\; y'(0)=2,\; y''(0)=-2,\; y'''(0)=0$$

[exer:9.2.24] $$y^{(4)}-6y'''+7y''+6y'-8y=0, \quad y(0)=-2,\quad y'(0)=-8,\quad y''(0)=-14$$,

$$y'''(0)=-62$$

[exer:9.2.25] $$4y^{(4)}-13y''+9y=0, \quad y(0)=1,\quad y'(0)=3,\quad y''(0)=1,\quad y'''(0)=3$$

[exer:9.2.26] $$y^{(4)}+2y'''-2y''-8y'-8y=0, \quad y(0)=5,\quad y'(0)=-2,\quad y''(0)=6,\quad y'''(0)=8$$

[exer:9.2.27] $$4y^{(4)}+8y'''+19y''+32y'+12y=0, \quad y(0)=3,\quad y'(0)=-3,\quad y''(0)= -7\over2$$, $$y'''(0)=31\over4}$$

[exer:9.2.28] 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. $$(D^2-2D+2)(D^2+1)y=0$$

[exer:9.2.29] $$(D^2+6D+13)(D-2)^2D^3y=0$$

[exer:9.2.30] $$(D-1)^2(2D-1)^3(D^2+1)y=0$$

[exer:9.2.31] $$(D^2+9)^3D^2y=0$$ [exer:9.2.32] $$(D-2)^3(D+1)^2Dy=0$$ [exer:9.2.33] $$(D^2+1)(D^2+9)^2(D-2)y=0$$ [exer:9.2.34] $$(D^4-16)^2y=0$$

[exer:9.2.35] $$(4D^2+4D+9)^3y=0$$

[exer:9.2.36] $$D^3(D-2)^2(D^2+4)^2y=0$$

[exer:9.2.37] $$(4D^2+1)^2(9D^2+4)^3y=0$$ [exer:9.2.38] $$\left[(D-1)^4-16\right]y=0$$

[exer:9.2.39] 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\le i 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 $$i<j$$.> Verify (A) for $$n=2$$ and $$n=3$$. Find the Wronskian of $$\{e^{{a_1}x}, \quad e^{{a_2}x},\dots, e^{{a_n}x}\}$$. [exer:9.2.40] 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.

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 $$y=0$$.

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 $$\{u_1,\dots,u_r,\allowbreak v_1,\dots,v_s\}$$ is a linearly independent set.

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.

[exer:9.2.41] 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 $$\le k-1$$.

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 $(D^2+\omega^2)^mz=0. \tag{B}$

Use Equation \ref{eq:9.2.17} to show that if $$y=e^{\lambda x}z$$ then $[(D-\lambda)^2+\omega^2]^my=e^{\lambda x}(D^2+\omega^2)^mz.\nonumber$ 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 $[(D-\lambda)^2+\omega^2]^my=0. \tag{C}$ Conclude from (d) that the functions $\begin{array}{rl} e^{\lambda x}\cos\omega x, xe^{\lambda x}\cos\omega x, &\dots, x^{m-1}e^{\lambda x}\cos\omega x,\\ e^{\lambda x}\sin\omega x, xe^{\lambda x}\sin\omega x,& \dots, x^{m-1}e^{\lambda x}\sin\omega x \end{array} \tag{\rm (D)}\nonumber$ are all solutions of (C).

Complete the proof of Theorem [thmtype:9.2.2} by showing that the functions in (D) are linearly independent.

[exer:9.2.42] Use the trigonometric identities \begin{aligned} \cos(A+B)&=&\cos A\cos B-\sin A\sin B\\ \sin(A+B)&=&\cos A\sin B+\sin A\cos B\end{aligned}\nonumber to show that $(\cos A+i\sin A)(\cos B+i\sin B)=\cos(A+B)+i\sin(A+B).\nonumber$

Apply (a) repeatedly to show that if $$n$$ is a positive integer then $\prod_{k=1}^n(\cos A_k+i\sin A_k)=\cos(A_1+A_2+\cdots+A_n) +i\sin(A_1+A_2+\cdots+A_n).\nonumber$

Infer from (b) that if $$n$$ is a positive integer then $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta. \tag{A}$

Show that (A) also holds if $$n=0$$ or a negative integer.

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 $$k$$?)

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 $z^n+\rho=(z-\rho^{1/n} \zeta_0)(z-\rho^{1/n} \zeta_1)\cdots(z-\rho^{1/n} \zeta_{n-1}).\nonumber$

[exer:9.2.43] Use (e) of Exercise [exer: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. $$y^{(5)}+y^{(4)}+y'''+y''+y'+y=0$$

[exer:9.2.44] 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 {dy\over dx}&=&\{dY\over dt}\\[4pt] \ x^2{d^2y\over dx^2}&=&\{d^2Y\over dt^2}-{dY\over dt}\\[4pt] \ x^3{d^3y\over dx^3}&=&\{d^3Y\over dt^3}-3{d^2Y\over dt^2}+2{dY\over 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 $$t$$.

[exer:9.2.45] Use Exercise [exer: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).