Multiple choice differential equation question.

Question

Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuously differentiable function satisfying $$y(t)=y(0)+\int_{0}^{t}y(s)ds,t\geq 0$$ Then

$1.y^{2}(t)=y^{2}(0)+\int_{0}^{t}y^{2}(s)ds$

$2.y^{2}(t)=y^{2}(0)+2\int_{0}^{t}y^{2}(s)ds$

$3.y^{2}(t)=y^{2}(0)+\int_{0}^{t}y(s)ds$

$4.y^{2}(t)=y^{2}(0)+(\int_{0}^{t}y(s)ds)^{2}+2y(0)\int_{0}^{t}y(s)$

Clearly last option is correct as its just square of the given $y(t).$ I am not getting any idea for other options how it comes square term inside the integration. Please help me. Thanks a lot.

Answer 1

Hint: $\frac{dy^2}{dt}=2y\frac{dy}{dt}$, so $y^2(t)$ can also be written as $y^2(0) + 2 \int_0^t y(s) y'(s) ds$. What is $y'$ in this case?