Summary separable differential equation

Separable Differential Equations
kosala Madhushanka
September 2024


1 Introduction
Separable Differential Equations definition
A differential equation is considered separable if it can be written in the
form:

dy f (x)
=
dx g(y)
Solution To solve a separable differential equation, we can follow these steps:
Separate the variables by multiplying both sides by dx and dividing both
sides by f (x)g(y). Integrate both sides with respect to their respective vari-
ables. If there are any given initial conditions, apply them to find the particular
solution. Initial Conditions Separable differential equations can be solved using
initial conditions in the following way:
Given an initial condition, say y(x0 ) = y0 , use it to find the constant resulting
from integration. Substitute this constant back into the general solution.
Example
Solve the following separable differential equation with the initial condition
y(1) = 2:

dy x2
= 3
dx y

y 3 dy = x2 dx
Z Z
y 3 dy = x2 dx + C

y4 x3
= +C
4 3
Applying the initial condition,

24 13
= +C
4 3


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