Differential equations — Abel's identity for the Wronskian

For $a\in\mathbb{R}$, let $y_1(x)$ and $y_2(x)$ be solutions of the differential equation

$$y"+\left(e^{x^2}+\cos x\right)y=0$$

such that $y_1(0)=3$, $y_1'(0)=-1$, $y_2(0)=-5$, $y_2'(0)=a$. Suppose $W(y_1,y_2)(x)$ denotes the Wronskian of $y_1$ and $y_2$. If $W(y_1,y_2)\left(\tfrac12\right)=4$, then the value of $a$ is