Question
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution to the differential equation is:
Solution
To solve the differential equation:
we can follow these steps:
1. Rewrite the Equation
First, let’s rewrite the equation in the standard linear form:
Divide both sides by
:
Here,
2. Find the Integrating Factor
The integrating factor
is given by:
3. Multiply Through by the Integrating Factor
Multiply both sides of the differential equation by
:
This simplifies to:
4. Integrate Both Sides
Integrate both sides with respect to
:
Let’s perform a substitution to evaluate the integral:
Let
, then
.
Notice that:
So the integral becomes:
5. Solve for
Substitute back to solve for
:
Divide both sides by
:
Final Solution
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg


The Deep Dive
This first-order linear differential equation can be rewritten in a more manageable form:
. The left side can be interpreted as a standard linear form, where the integrating factor would be
. Multiply through by this integrating factor to solve for
.
In the realm of real-world applications, differential equations like this often model phenomena in physics and engineering. For instance, they can represent the rate of change of population in ecology, electrical circuits, or even the motion of a pendulum. Understanding how to tackle these equations is crucial for predicting and controlling such systems effectively!