Question #311026

Solve for x in the following set of simultaneous differential equations by using D-operator methods:

(D + 2) x - 3y = 1

-3x + (D + ) y = e-t

1
Expert's answer
2022-03-15T09:20:11-0400

Given the simultaneous differential equations,

(D+2)x3y=1(1)3x+(D+2)y=et (2)\begin{aligned} (D + 2) x - 3y &= 1\qquad(1)\\ -3x + (D + 2) y &= e^{-t} ~\quad(2) \end{aligned}


Applying (D+2)(D + 2) operator for (1), Multiplying (2) by 3 and adding we get,


(D+2)2x3(D+2)y=2(1)9x+3(D+2)y=3et   (2)(D+2)2x9x=2+3et(D2+4D+49)x=2+3et(D2+4D5)x=2+3et\begin{aligned} (D + 2)^2 x - 3(D+2)y &= 2\qquad(1)\\ -9x + 3(D + 2) y &= 3e^{-t} ~~~(2)\\ \hline (D + 2)^2 x -9x &= 2+3e^{-t}\\ (D^2 + 4D + 4 - 9)x & = 2+3e^{-t}\\ (D^2 + 4D -5)x & = 2+3e^{-t}\\ \end{aligned}

The auxiliary equation is, m2+4m5=0.m^{2} + 4m - 5 = 0. Solving we get m=5,1m = -5,1.

The complementary function for x(t) is, C.F=c1e5t+c2et\text{C.F} = c_1e^{-5t} + c_2 e^{t}

Particular Integral for x(t)=1D2+4D5(2+3et)=253et8(Replacing D by 0 and -1 for first and second term respectively)\begin{aligned} \text{Particular Integral for }x(t) &= \frac{1}{D^{2}+4D-5} (2+3e^{-t})\\ &=-\frac{2}{5} -\frac{3e^{-t}}{8}\quad(\text{Replacing D by 0 and -1 for first} \\ &\qquad \qquad \qquad \qquad\text{ and second term respectively})\\ \end{aligned}

x(t)=c1e5t+c2et(25+3et8)\therefore x(t) = c_{1}e^{-5t}+c_2e^t-\Big(\dfrac{2}{5}+\dfrac{3e^{-t}}{8}\Big)


From equation (1), 3y=Dx+2x13y = Dx+2x -1


3y=D(c1e5t+c2et(25+3et8))+2(c1e5t+c2et(25+3et8))13y=5c1e5t+c2et+3et8+2c1e5t+2c2et45+3et413y=3c1e5t+3c2et3et895y(t)=c1e5t+c2etet835\begin{aligned} 3y&=D\Bigg(c_{1}e^{-5t}+c_2e^t-\Big(\dfrac{2}{5}+\dfrac{3e^{-t}}{8}\Big)\Bigg) + 2\Bigg(c_{1}e^{-5t}+c_2e^t-\Big(\dfrac{2}{5}+\dfrac{3e^{-t}}{8}\Big)\Bigg) - 1\\ 3y &= -5c_{1}e^{-5t}+c_2e^t+\dfrac{3e^{-t}}{8} + 2c_{1}e^{-5t}+2c_2e^t-\dfrac{4}{5}+\dfrac{3e^{-t}}{4}- 1\\ 3y &= -3c_{1}e^{-5t}+3c_2e^t-\dfrac{3e^{-t}}{8} -\dfrac{9}{5}\\ \therefore y(t)& = -c_{1}e^{-5t}+c_2e^t-\dfrac{e^{-t}}{8} -\dfrac{3}{5}\\ \end{aligned}


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