Find a solution to \(\displaystyle{x}'{\left({t}\right)}+\int^{{{t}}}_{\left\lbrace{0}\right\rbrace}{\left({t}-{s}\right)}{x}{\left({s}\right)}{d}{s}={t}+{\frac{{{1}}}{{{2}}}}{t}^{{2}}+{\frac{{{1}}}{{{24}}}}{t}^{{4}}\)

Erik Cantu

Erik Cantu

Answered question

2022-04-05

Find a solution to x(t)+{0}t(ts)x(s)ds=t+12t2+124t4

Answer & Explanation

Nunnaxf18

Nunnaxf18

Beginner2022-04-06Added 18 answers

Taking the Laplace transform of the left hand side, using the convolution formula on the integral (note the integral is f⋆g where f is the identity function and and g=x):
L(x(t)+{0}t(ts)x(s)ds)=sX(s)x(0)+1s2X(s).
Taking the Laplace transform of the right hand side:
L(t+12t2+124t4)=1s2+122s3+124·24s5.
So, we have:
sX(s)-x(0)+1s2·X(s)=1s2+1s3+1s5
Now solve for X(s) and then take the inverse transform to find x(t).

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