Question

Let $f: R \rightarrow R$ be a differentiable function such that $f^{\prime}(x)+f(x)=\int\limits_0^2 f(t) d t$If $f(0)=e^{-2}$, then $2 f(0)-f(2)$ is equal to_____

Answer 1

The correct answer is 1.
dxdy​+y=k
y⋅ex=k⋅ex+c
f(0)=e−2
⇒c=e−2−k
∴y=k+(e−2−k)e−x
now k=0∫2​(k+(e−2−k)e−x)dx
⇒k=e−2−1
∴y=(e−2−1)+e−x
f(2)=2e−2−1,f(0)=e−2
2f(0)−f(2)=1