$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 9.3E: Undetermined Coefficients for Higher Order Equations (Exercises)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

In Exercises [exer:9.3.1}– [exer:9.3.59} find a particular solution.

[exer:9.3.1] $$y'''-6y''+11y'-6y=-e^{-x}(4+76x-24x^2)$$

[exer:9.3.2] $$y'''-2y''-5y'+6y=e^{-3x}(32-23x+6x^2)$$

[exer:9.3.3] $$4y'''+8y''-y'-2y=-e^x(4+45x+9x^2)$$

[exer:9.3.4] $$y'''+3y''-y'-3y=e^{-2x}(2-17x+3x^2)$$

[exer:9.3.5] $$y'''+3y''-y'-3y=e^x(-1+2x+24x^2+16x^3)$$

[exer:9.3.6] $$y'''+y''-2y=e^x(14+34x+15x^2)$$

[exer:9.3.7] $$4y'''+8y''-y'-2y=-e^{-2x}(1-15x)$$

[exer:9.3.8] $$y'''-y''-y'+y=e^x(7+6x)$$

[exer:9.3.9] $$2y'''-7y''+4y'+4y=e^{2x}(17+30x)$$

[exer:9.3.10] $$y'''-5y''+3y'+9y=2e^{3x}(11-24x^2)$$

[exer:9.3.11] $$y'''-7y''+8y'+16y=2e^{4x}(13+15x)$$

[exer:9.3.12] $$8y'''-12y''+6y'-y=e^{x/2}(1+4x)$$

[exer:9.3.13] $$y^{(4)}+3y'''-3y''-7y'+6y=-e^{-x}(12+8x-8x^2)$$

[exer:9.3.14] $$y^{(4)}+3y'''+y''-3y'-2y=-3e^{2x}(11+12x)$$

[exer:9.3.15] $$y^{(4)}+8y'''+24y''+32y'=-16e^{-2x}(1+x+x^2-x^3)$$

[exer:9.3.16] $$4y^{(4)}-11y''-9y'-2y=-e^x(1-6x)$$

[exer:9.3.17] $$y^{(4)}-2y'''+3y'-y=e^x(3+4x+x^2)$$

[exer:9.3.18] $$y^{(4)}-4y'''+6y''-4y'+2y=e^{2x}(24+x+x^4)$$

[exer:9.3.19] $$2y^{(4)}+5y'''-5y'-2y=18e^x(5+2x)$$

[exer:9.3.20] $$y^{(4)}+y'''-2y''-6y'-4y=-e^{2x}(4+28x+15x^2)$$

[exer:9.3.21] $$2y^{(4)}+y'''-2y'-y=3e^{-x/2}(1-6x)$$

[exer:9.3.22] $$y^{(4)}-5y''+4y=e^x(3+x-3x^2)$$

[exer:9.3.23] $$y^{(4)}-2y'''-3y''+4y'+4y=e^{2x}(13+33x+18x^2)$$

[exer:9.3.24] $$y^{(4)}-3y'''+4y'=e^{2x}(15+26x+12x^2)$$

[exer:9.3.25] $$y^{(4)}-2y'''+2y'-y=e^x(1+x)$$

[exer:9.3.26] $$2y^{(4)}-5y'''+3y''+y'-y=e^x(11+12x)$$

[exer:9.3.27] $$y^{(4)}+3y'''+3y''+y'=e^{-x}(5-24x+10x^2)$$

[exer:9.3.28] $$y^{(4)}-7y'''+18y''-20y'+8y=e^{2x}(3-8x-5x^2)$$

[exer:9.3.29] $$y'''-y''-4y'+4y=e^{-x}\left[(16+10x)\cos x+(30-10x)\sin x\right]$$

[exer:9.3.30] $$y'''+y''-4y'-4y=e^{-x}\left[(1-22x)\cos 2x-(1+6x)\sin2x\right]$$

[exer:9.3.31] $$y'''-y''+2y'-2y=e^{2x}[(27+5x-x^2)\cos x+(2+13x+9x^2)\sin x]$$

[exer:9.3.32] $$y'''-2y''+y'-2y=-e^x[(9-5x+4x^2)\cos 2x-(6-5x-3x^2)\sin2x]$$

[exer:9.3.33] $$y'''+3y''+4y'+12y=8\cos2x-16\sin2x$$

[exer:9.3.34] $$y'''-y''+2y=e^x[(20+4x)\cos x-(12+12x)\sin x]$$

[exer:9.3.35] $$y'''-7y''+20y'-24y=-e^{2x}[(13-8x)\cos 2x-(8-4x)\sin2x]$$

[exer:9.3.36] $$y'''-6y''+18y'=-e^{3x}[(2-3x)\cos 3x-(3+3x)\sin3x]$$

[exer:9.3.37] $$y^{(4)}+2y'''-2y''-8y'-8y=e^x(8\cos x+16\sin x)$$

[exer:9.3.38] $$y^{(4)}-3y'''+2y''+2y'-4y=e^x(2\cos2x -\sin2x)$$

[exer:9.3.39] $$y^{(4)}-8y'''+24y''-32y'+15y=e^{2x}(15x\cos2x+32\sin2x)$$

[exer:9.3.40] $$y^{(4)}+6y'''+13y''+12y'+4y=e^{-x}[(4-x)\cos x-(5+x)\sin x]$$

[exer:9.3.41] $$y^{(4)}+3y'''+2y''-2y'-4y=-e^{-x} (\cos x-\sin x)$$

[exer:9.3.42] $$y^{(4)}-5y'''+13y''-19y'+10y=e^x (\cos2x+\sin2x)$$

[exer:9.3.43] $$y^{(4)}+8y'''+32y''+64y'+39y=e^{-2x}[(4-15x)\cos3x-(4+15x)\sin 3x]$$

[exer:9.3.44] $$y^{(4)}-5y'''+13y''-19y'+10y=e^x[(7+8x)\cos 2x+(8-4x)\sin2x]$$

[exer:9.3.45] $$y^{(4)}+4y'''+8y''+8y'+4y=-2e^{-x} (\cos x-2\sin x)$$

[exer:9.3.46] $$y^{(4)}-8y'''+32y''-64y'+64y=e^{2x} (\cos2x-\sin2x)$$

[exer:9.3.47] $$y^{(4)}-8y'''+26y''-40y'+25y=e^{2x}[3\cos x-(1+3x)\sin x]$$

[exer:9.3.48] $$y'''-4y''+5y'-2y=e^{2x}-4e^x-2\cos x+4\sin x$$

[exer:9.3.49] $$y'''-y''+y'-y=5e^{2x}+2e^x-4\cos x+4\sin x$$

[exer:9.3.50] $$y'''-y'=-2(1+x)+4e^x-6e^{-x}+96e^{3x}$$

[exer:9.3.51] $$y'''-4y''+9y'-10y=10e^{2x}+20e^x\sin2x-10$$

[exer:9.3.52] $$y'''+3y''+3y'+y=12e^{-x}+9\cos2x-13\sin2x$$

[exer:9.3.53] $$y'''+y''-y'-y=4e^{-x}(1-6x)-2x\cos x+2(1+x)\sin x$$

[exer:9.3.54] $$y^{(4)}-5y''+4y=-12e^x+6e^{-x}+10\cos x$$

[exer:9.3.55] $$y^{(4)}-4y'''+11y''-14y'+10y=-e^x(\sin x+2\cos2x)$$

[exer:9.3.56] $$y^{(4)}+2y'''-3y''-4y'+4y=2e^x(1+x)+e^{-2x}$$

[exer:9.3.57] $$y^{(4)}+4y=\sinh x\cos x-\cosh x\sin x$$

[exer:9.3.58] $$y^{(4)}+5y'''+9y''+7y'+2y=e^{-x}(30+24x)-e^{-2x}$$

[exer:9.3.59] $$y^{(4)}-4y'''+7y''-6y'+2y=e^x(12x-2\cos x+2\sin x)$$

[exer:9.3.60] $$y'''-y''-y'+y=e^{2x}(10+3x)$$

[exer:9.3.61] $$y'''+y''-2y=-e^{3x}(9+67x+17x^2)$$

[exer:9.3.62] $$y'''-6y''+11y'-6y=e^{2x}(5-4x-3x^2)$$

[exer:9.3.63] $$y'''+2y''+y'=-2e^{-x}(7-18x+6x^2)$$

[exer:9.3.64] $$y'''-3y''+3y'-y=e^x(1+x)$$

[exer:9.3.65] $$y^{(4)}-2y''+y=-e^{-x}(4-9x+3x^2)$$

[exer:9.3.66] $$y'''+2y''-y'-2y=e^{-2x}\left[(23-2x)\cos x+(8-9x)\sin x\right]$$

[exer:9.3.67] $$y^{(4)}-3y'''+4y''-2y'=e^x\left[(28+6x)\cos 2x+(11-12x)\sin2x\right]$$

[exer:9.3.68] $$y^{(4)}-4y'''+14y''-20y'+25y=e^x\left[(2+6x)\cos 2x+3\sin2x\right]$$

[exer:9.3.69] $$y'''-2y''-5y'+6y=2e^x(1-6x),\quad y(0)=2, \quad y'(0)=7,\quad y''(0)=9$$

[exer:9.3.70] $$y'''-y''-y'+y=-e^{-x}(4-8x),\quad y(0)=2, \quad y'(0)=0,\quad y''(0)=0$$

[exer:9.3.71] $$4y'''-3y'-y=e^{-x/2}(2-3x),\quad y(0)=-1, \quad y'(0)=15,\quad y''(0)=-17$$

[exer:9.3.72] $$y^{(4)}+2y'''+2y''+2y'+y=e^{-x}(20-12x),\, y(0)=3,\; y'(0)=-4,\; y''(0)=7,\; y'''(0)=-22$$

[exer:9.3.73] $$y'''+2y''+y'+2y=30\cos x-10\sin x, \quad y(0)=3,\quad y'(0)=-4,\quad y''(0)=16$$

[exer:9.3.74] $$y^{(4)}-3y'''+5y''-2y'=-2e^x(\cos x-\sin x),\; y(0)=2,\; y'(0)=0,\; y''(0)~=~-1, \; y'''(0)=-5$$

[exer:9.3.75] Prove: A function $$y$$ is a solution of the constant coefficient nonhomogeneous equation

$a_0y^{(n)}+a_1y^{(n-1)}+\cdots+a_ny=e^{\alpha x}G(x) \tag{A}$

if and only if $$y=ue^{\alpha x}$$, where $$u$$ satisfies the differential equation

$a_0u^{(n)}+{p^{(n-1)}(\alpha)\over(n-1)!}u^{(n-1)}+ {p^{(n-2)}(\alpha)\over(n-2)!}u^{(n-2)}+\cdots+p(\alpha)u=G(x) \tag{B}$

and

$p(r)=a_0r^n+a_1r^{n-1} + \cdots + a_n\nonumber$

is the characteristic polynomial of the complementary equation

$a_0y^{(n)}+a_1y^{(n-1)}+\cdots+a_ny=0.\nonumber$

[exer:9.3.76] Prove:

The equation

$\begin{array}{lcl} a_0u^{(n)}&+&\{p^{(n-1)}(\alpha)\over(n-1)!}u^{(n-1)}+ \{p^{(n-2)}(\alpha)\over(n-2)!}u^{(n-2)}+\cdots+p(\alpha)u\\ &=&\left(p_0+p_1x+\cdots+p_kx^k\right)\cos \omega x\\&&\,+ \left(q_0+q_1x+\cdots+q_kx^k\right)\sin\omega x \end{array} \tag{A}$

has a particular solution of the form

$u_p=x^m\left(u_0+u_1x+\cdots+u_kx^k\right)\cos\omega x+ \left(v_0+v_1x+\cdots+v_kx^k\right)\sin\omega x.\nonumber$

If $$\lambda+i\omega$$ is a zero of $$p$$ with multiplicity $$m\ge1$$, then (A) can be written as

$a(u''+\omega^2 u)= \left(p_0+p_1x+\cdots+p_kx^k\right)\cos\omega x+ \left(q_0+q_1x+\cdots+q_kx^k\right)\sin\omega x,\nonumber$

which has a particular solution of the form

$u_p=U(x)\cos\omega x+V(x)\sin\omega x,\nonumber$

where

$U(x)=u_0x+u_1x^2+\cdots+u_kx^{k+1},\,V(x)=v_0x+v_1x^2+\cdots+v_kx^{k+1}\nonumber$

and

$\begin{array}{rcl} a(U''(x)+2\omega V'(x))&=&p_0+p_1x+\cdots+p_kx^k\\[10pt] a(V''(x)-2\omega U'(x))&=&q_0+q_1x+\cdots+q_kx^k. \end{array}\nonumber$