Table of Contents

Solving Separable Differential Equations Examples 4
Recall from the Solving Separable Differential Equations page that if we have a separable differential equation $M(x) + N(y) \frac{dy}{dx} = 0$, then we can rewrite it as:
(1)
Provided that these integrals can be evaluated and that they’re not too difficult to do so, then we can obtain solutions for the separable differential equation.
We will now look at some examples of solving separable differential equations. In these examples, we will concern ourselves with determining the Interval of Validity, which is the largest interval for which our solution is valid that contains the initial condition given.
Example 1
Solve the differential equation $\frac{dy}{dt} = \frac{y^2 + 1}{t + 1}$.
This differential equation can be rewritten and solved for as:
(2)
Example 2
Solve the initial value problem $\frac{dy}{dt} = \frac{4 \sin 2t}{y}$ with the initial condition $y(0) = 1$.
The given differential equation is separable, and can be rewritten and solved for as:
(3)
From the initial value problem, we have that $y(0) = 1$ implies that we take the positive square root from above. Plugging in the initial condition and we have that:
(4)
Therefore the solution to our initial value problem is:
(5)
Example 3
Solve the differential equation $\frac{dy}{dx} = \frac{y^2(x – 3)}{x^3}$.
This differential equation is separable and can be rewritten and solved as follows:
(6)
Related post:
 New technique developed to detect autism in children – EurekAlert
 Gujarat Board to offer Mathematics to NonScience Students from this year onwards – Jagran Josh
 Institute of Mathematics & Application (IMA) Recruitment 2019 for Professor Posts – Jagran Josh
 Help with primary school mathematics — September is just nine weeks away – Galway Advertiser
 Government Launches Effort to Strengthen Math Skills & Improve Job Prospects – Government of Ontario News
 Chaiwalla to Doctor: 5 Teachers Providing Free JEE/NEET Coaching to Needy Students – The Better India
 A celebration of Science, Technology, Engineering, and Mathematics (STEM) – Daily Trust
 Sum of a life – THE WEEK
 Assistant Professor (Tenure Track) in Mathematics in South Holland, Delft – IamExpat in the Netherlands
 Standing in Galileo’s shadow: Why Thomas Harriot should take his place in the scientific hall of fame – OUPblog