Separable Differential Equations

Note: $\displaystyle \exp(x) = e^x$

First, we separate the equations, by turning $\displaystyle \frac{dy}{dx} = \frac{\exp(x)}{4y}$ into $\displaystyle \int 4y dy = \int \exp(x) dx$ by multiplying and taking the integral of both sides. Next, we evaluate the integral, getting $\displaystyle 2y^2 = \exp(x) + C$ , as we only need one constant. You could put the constant on the other side as well, but it wouldn't make a difference. A constant minus a constant is a constant. To solve for C, let's plug in $\displaystyle y(0) = 2$ to get

$\displaystyle 2(2)^2 = \exp(0) + C$

$\displaystyle 8 = 1 + C$

$\displaystyle C = 7$

Plugging in C, we get $\displaystyle 2y^2 = \exp(x) + 7$ , and we divide by 2 and take the square root of both sides to get $\boxed{\sqrt{\frac{\exp(x) + 7}{2}}}$