# Integral equations

Why Mathehtica could not solve the simple equation $$\text{DSolve}\left[-\int_0^x t^4 (t-x)^4 y(t) \, dt-\frac{313 x^4}{15120}+\frac{115 x^2}{252}+\left(\frac{13 x^4}{15120}+\frac{11 x^2}{252}+1\right) \cos (x)-1=0,y(x),x\right]$$ Mathematica it is capable to solve other more difficult equation why fail in this simple case thanks it is possible get a solution derivating thanks

• DSolve[]'s support for integral equations is still somewhat limited, so don't be surprised if some things don't work yet. – Mariusz Iwaniuk Nov 17 '17 at 11:36
• Thanks Mariusz Iwaniuk – antonio asis Nov 17 '17 at 11:50

eq = Inactivate[-Integrate[t^4*(t - x)^4*y[t], {t, 0, x}] -
313*x^4/15120 + 115*x^2/252 + (13*x^4/15120 + 11 x^2/252 + 1)*Cos[x] - 1 == 0, Integrate]


Differentiate five times the eq give a solution:

sol = First@Solve[D[Activate@eq, {x, 5}], y[x]] // Expand


$\left\{y(x)\to -\frac{29 \sin (x)}{3024 x^4}+\frac{29 \cos (x)}{3024 x^3}+\frac{5 \sin (x)}{2016 x^2}-\frac{13 \sin (x)}{362880}+\frac{13 \cos (x)}{18144 x}\right\}$

and check if everything is fine:

Activate@eq /. y[t] -> (y[x] /. sol[] /. x -> t) // FullSimplify

(* True *)