y ′ ′ + 3 y − 2 y = e − x x . This is a second order non-homogeneous DE The characteristics equation is ; m 2 + 3 m − 2 = 0 m = − 3 ± 3 2 − 4 ( 1 ) ( − 2 ) 2 ( 1 ) m = − 3 ± 17 2 The complimentary solution is; y c = A e − 3 + 17 2 x + B e − 3 − 17 2 x . To get he Particular solution is; y p = C 1 y 1 + C 2 y 2 Where: C 1 = ∫ − y 2 f ( x ) W ( x ) and C 2 = ∫ y 1 f ( x ) W ( x ) W ( x ) = y 1 y 2 ′ − y 1 ′ y 2 W ( x ) = ( − 3 − 17 2 x ) e − 3 + 17 2 x e − 3 − 17 2 x − ( − 3 + 17 2 x ) e − 3 + 17 2 x e − 3 − 17 2 x = − 17 e − 3 x f ( x ) = e − x x C 1 = ∫ e − 3 + 17 2 x x 17 d x = Ei ( 1 − 17 2 x ) 17 + C C 2 = ∫ e 7 − 17 2 x − x 17 d x = − Ei ( 7 − 17 2 x ) 17 + C The general solution is given as; y ( x ) = A y 1 + B y 2 + C 1 y 1 + C 2 y 2 y ( x ) = A e − 3 + 17 2 x + B e − 3 − 17 2 x + Ei ( 1 − 17 2 x ) 17 e − 3 + 17 2 x − Ei ( 7 − 17 2 x ) 17 e − 3 − 17 2 x . y'' + 3y - 2y = \frac{ e^{-x}}{x}.\\
\text{This is a second order non-homogeneous DE}\\
\text{The characteristics equation is ;}\\
m^2+3m-2=0\\
m=\frac{-3 \pm \sqrt{3^2-4(1)(-2)}}{2(1)}\\
m=\frac{-3 \pm \sqrt{17}}{2}\\
\text{The complimentary solution is;}\\
y_c= Ae^{\frac{-3 + \sqrt{17}}{2}x} + Be^{\frac{-3 - \sqrt{17}}{2}x}.\\
\text{To get he Particular solution is;}\\
y_p=C_1y_1+C_2y_2\\
\text{Where:} ~~~~C_1=\int \frac{-y_2f(x)}{W(x)} ~~ \text{and}~~ C_2=\int \frac{y_1f(x)}{W(x)}\\
W(x)=y_1y_2'-y_1'y_2\\
W(x)=\left(\frac{-3 - \sqrt{17}}{2}x\right)e^{\frac{-3 + \sqrt{17}}{2}x}e^{\frac{-3 - \sqrt{17}}{2}x}-\left(\frac{-3 + \sqrt{17}}{2}x\right)e^{\frac{-3 + \sqrt{17}}{2}x}e^{\frac{-3 - \sqrt{17}}{2}x}=-\sqrt{17} e^{-3x}\\\\
f(x)=\frac{e^{-x}}{x}\\\\
\\C_1=\int\frac{e^{\frac{-3 + \sqrt{17}}{2}x}}{x\sqrt{17}}dx=\frac{\operatorname{Ei}{\left(\frac{ 1 - \sqrt{17}}{2}x \right)}}{\sqrt{17}}+C\\
C_2=\int\frac{e^{\frac{7 - \sqrt{17}}{2}x}}{-x\sqrt{17}}dx=-\frac{\operatorname{Ei}{\left(\frac{ 7- \sqrt{17}}{2}x \right)}}{\sqrt{17}}+C\\\\
\text{The general solution is given as;}\\
y(x)=Ay_1+By_2+C_1y_1+C_2y_2\\\\
y(x)=Ae^{\frac{-3 + \sqrt{17}}{2}x} + Be^{\frac{-3 - \sqrt{17}}{2}x}+\frac{\operatorname{Ei}{\left(\frac{ 1 - \sqrt{17}}{2}x \right)}}{\sqrt{17}}e^{\frac{-3 + \sqrt{17}}{2}x}-\frac{\operatorname{Ei}{\left(\frac{ 7- \sqrt{17}}{2}x \right)}}{\sqrt{17}}e^{\frac{-3 - \sqrt{17}}{2}x}. y ′′ + 3 y − 2 y = x e − x . This is a second order non-homogeneous DE The characteristics equation is ; m 2 + 3 m − 2 = 0 m = 2 ( 1 ) − 3 ± 3 2 − 4 ( 1 ) ( − 2 ) m = 2 − 3 ± 17 The complimentary solution is; y c = A e 2 − 3 + 17 x + B e 2 − 3 − 17 x . To get he Particular solution is; y p = C 1 y 1 + C 2 y 2 Where: C 1 = ∫ W ( x ) − y 2 f ( x ) and C 2 = ∫ W ( x ) y 1 f ( x ) W ( x ) = y 1 y 2 ′ − y 1 ′ y 2 W ( x ) = ( 2 − 3 − 17 x ) e 2 − 3 + 17 x e 2 − 3 − 17 x − ( 2 − 3 + 17 x ) e 2 − 3 + 17 x e 2 − 3 − 17 x = − 17 e − 3 x f ( x ) = x e − x C 1 = ∫ x 17 e 2 − 3 + 17 x d x = 17 Ei ( 2 1 − 17 x ) + C C 2 = ∫ − x 17 e 2 7 − 17 x d x = − 17 Ei ( 2 7 − 17 x ) + C The general solution is given as; y ( x ) = A y 1 + B y 2 + C 1 y 1 + C 2 y 2 y ( x ) = A e 2 − 3 + 17 x + B e 2 − 3 − 17 x + 17 Ei ( 2 1 − 17 x ) e 2 − 3 + 17 x − 17 Ei ( 2 7 − 17 x ) e 2 − 3 − 17 x .
#331389 on Nov 2022
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