Question

Solve the integral equation \[y(t)=t+\int^t_0 y(x)\,dx+\int^t_0 (t-x)y(x)\,dx\]

Answer 1

Using Laplace Transform i think ha?

Answer 2

yup for sure

Answer 3

actually I might go to bed haha. don't want to close this question though as I will be trying again in the morning. some tips would be appreciated. I'm just not sure If I have to evaluate the integrals

Answer 4

suppose that \(\mathscr{L} [f(x)]=F(s) \) and \(\mathscr{L} [g(x)]=G(s) \)

just note that

\[\mathscr{L} \left( \int\limits_{0}^{t} f(x) dx\right)=\frac{F(s)}{s} \\ \]
and
\[\mathscr{L} \left( \int\limits_{0}^{t} f(x) g(t-x)dx\right)=F(s) G(s) \\ \]

Answer 5

here u have \(g(x)=x\)

Answer 6

the latter part is convolution ...i guess

Answer 7

so after applying laplace for equation i think we have :

\[Y(s)=\frac{1}{s^2}+\frac{Y(s)}{s}+\frac{Y(s)}{s^2}\]

Answer 8

and

\[Y(s)=\frac{1}{s^2-s-1}\]

Answer 9

Wait ...