Solve the integral equation y(x)=displaystyle∈t_0^x y(t)…

Solve the integral equation $y(x)=\displaystyle\int_0^x y(t)\,dt+x+1$.

Explanation of Method

Differentiate both sides to convert the integral equation into an initial value problem for a linear ODE, using the original equation at $x=0$ to recover the missing initial condition.

Step 1: Differentiate

By the Fundamental Theorem of Calculus, differentiating $y(x)=\displaystyle\int_0^x y(t)\,dt+x+1$ gives

\[y'=y+1,\qquad\text{i.e.}\qquad y'-y=1.\]

Step 2: Solve the linear equation

With integrating factor $e^{-x}$: $\dfrac{d}{dx}(e^{-x}y)=e^{-x}$, so $e^{-x}y=-e^{-x}+C$, giving $y=Ce^{x}-1$.

Step 3: Recover the initial condition

Setting $x=0$ in the original integral equation: $y(0)=\displaystyle\int_0^0 y\,dt+0+1=1$. Then $y(0)=C-1=1\Rightarrow C=2$.

Result

Final answer:

\[\boxed{\,y(x)=2e^{x}-1.}\]