"Solution"
To evaluate the particular solution "l[y]=x^2" .
It was given that "\\sin(x)" is a solution of
Next,
For D.E "\\frac{d^4y}{dx^4}+2\\frac{d^3y}{dx^3}+6\\frac{d^2y}{dx^2}+2\\frac{dy}{dx}+5y=x^2" , you need to try the solution
"y_{p_1}=Ax^2+Bx+C" (Since "l[y]=x^2" is of degree 2), Such that
"6A+2B+5(Ax^2+Bx+c)=x^2"
"5Ax^2+6A+5Bx+2B+5C=x^2"
Equating the coefficients of like powers yields:
"5B=0 \\implies B=0"
"6A+2B+5C=0 \\implies 6A+5C=0 \\implies 5C=-\\frac{6}{5} \\implies C=-\\frac{6}{25}"
Hence, evaluating "y_{p_1}" yields
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