Question

Non-Homogenous Linear Differential Equations

1. (2D ^ 2 + 3D + 1) * y = e ^ (- 3x)

2.(D^ 2 -2D+5)y=25x^ 2 +12

4. (D ^ 2 - 3D + 2) * y = 14sin 2x - 18cos 2x

Answer 1

2D² + 3D + 1)$\times$ y =e^{(- 3x)}

(D+1)(2D+1)=e^{(- 3x)}

(D+1)(2D+1)=$\frac{e}{3x}$

D=3, $\frac{1+\sqrt(2-3)}{3}$

y(x)= $\frac{1}{D^-2D^3+D-3}$ ex

y(x)= $\frac{1}{1-2+1-2}$ ex

y= $\frac{1}{1-2+1-2}$ x^ex

(D² -2D+5)y=25x² +12

(D-1)(D-5)y=25x²+12

y= $\frac{25x2+12}{(D-1)(D-5)}$

D=1, $\frac{1+\sqrt(25+12)}{5}$

y(x)=$\frac{6.08}{D^-2D^5+D-3}$

(D² - 3D + 2)$\times$ y = 14sin2x-18cos 2x

(D-1)(D+2)y=14sin2x-18cos2x

D=1, $\frac{sin2+\sqrt(14-2)}{5}$

y=$\frac{(14sin^2(x))}{(d-1)(d+2)}$ -$\frac{(18cos^2(x)}{(d-1)(d+2)}$