You may recall that the inverse of a derivative is the integral. For some differential equations, we can solve simply by integrating both sides, as long as each side only has one variable. These type of differential equations are called separable. A first-order separable differential equation can be written in the form:
\( \cfrac{dy}{dx} = g(x)f(y)\)
\( M(x)dx + N(y)dy =0\)
We can then separate the functions, integrate on both sides and isolate for \(y\). A simple example is to consider the DE \(\cfrac{dy}{dx} = y\). We know the solution to this equation is \(y = e^x\) since the derivate is itself.
\( \cfrac{dy}{dx} = y\)
\( \cfrac{dy}{y} = dx\)
\( \int {\cfrac{dy}{y}} = \int {dx}\)
\( \ln{y} + C_1 = x + C_2\)
\( \ln{y} = x + C_3\)
\( e^{\ln{y}} = e^{x + C_3}\)
\( y = e^{c_3}e^{x}\)
\( y = Ce^{x}\)
Here, we could take \(C = 1\) for the simplest solution or use the initial value to solve for the particular solution.
Is the equation \(y^{'} - (x+y) \sin {x} = 0\) separable?
Solve the equation \(\cfrac{dy}{dx} = \cfrac{2x^2 - 1}{y+1} \) given \(y(0)=3\).