Variable Separable Differential Equation

Rarely Tested

Variable Separable Differential Equation

A differential equation is separable if it can be written as:

$$\frac{dy}{dx}=f(x)g(y)$$

Then:

$$\frac{dy}{g(y)}=f(x)\,dx$$

Usage

- Used when all terms involving $$y$$ can be placed on one side and all terms involving $$x$$ on the other.

Question 1

Let $$y:(-\infty,\infty)\to(0,\infty)$$ be the solution of the differential equation

$$\dfrac{dy}{dx}=\dfrac{e^{5x}y^3+y^3}{e^x+e^x y^4},$$

satisfying $$y(0)=\dfrac{1}{\sqrt{2}}$$. Then the value of $$y(\log_e 2)$$ is

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