Answer to Question #223870 in Chemical Engineering for Lokika

Question #223870

Solve d^{2}y/dx^{2}-6dy/dx+13y=20e^{3x}cos(1x+5)


1
Expert's answer
2021-08-26T01:05:04-0400

Given DE:


"y''(x)-6y'(x)+13y(x)=20 e^{3x}\\cos (x+5)"

The characteristic equation for homogeneous DE


"k^2-6k+13=0"

Roots:

"k_1=3-2i,\\quad k_2=3+2i"

Thus, the general solution of homogeneous DE

"y_h(x)=e^{3x}(C_1 \\cos 2x+C_2\\sin 2x)"

The partial solution of inhomogeneous DE

"y_i(x)=\\frac{20}{3}e^{3x}\\cos(x+5)"

Finally,

"y(x)=y_h(x)+y_i(x)=\\\\\n=e^{3x}(C_1 \\cos 2x+C_2\\sin 2x)+\\frac{20}{3}e^{3x}\\cos(x+5)"


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