Solving the equation $\int_0^t \frac{y(x)}{\sqrt{t-x}}dx=1+t+t^2$ by Laplace transform: Why does convolution work, but not my method?

Question

Question:

Solve the equation $\int_0^t \frac{y(x)}{\sqrt{t-x}}dx=1+t+t^2$

My try:

I applied Laplace transform to both sides of the equation.

$L\{\int_0^t \frac{y(x)}{\sqrt{t-x}}dx\} = \frac{1}{s}+\frac{1}{s^2}+\frac{2}{s^3}$

The problem is about $L\{\int_0^t \frac{y(x)}{\sqrt{t-x}}dx\} $. We know that $L\{\int_0^tf(x)dx\}=\frac{1}{s}L\{f(t)\}$. So, we have:

$L\{\int_0^t \frac{y(x)}{\sqrt{t-x}}dx\} =\frac{1}{s}L\{\frac{y(t)}{\sqrt{t-t}}\}=\frac{1}{s}L\{\frac{y(t)}{\sqrt{0}}\}$

As you see, I get $\sqrt{0}$ as a part of the solution!

What's wrong? How should I fix this?

Edit: I know the convolution works for this equation. But the other way (the formula I mentioned above) should work too! doesn't it? What is wrong with my way?

Answer 1

$L\{ \int_0^t y(x) (t-x)^{-1/2}dx\} = L\{y(t)\} \times L\{t^{-1/2} \} = Y(s) \times \frac{\Gamma(-1/2)}{s^{1/2}}$