https://file.aiquickdraw.com/mathbot/file/1729466229726o5zizhbt.jpg

The given differential equation is:

\[
y' = 5xy^2
\]

This is a separable differential equation, meaning we can solve it by separating the variables and then integrating both sides.

### Steps to solve:

1. **Separate variables**:  
   \[
   \frac{dy}{y^2} = 5x \, dx
   \]

2. **Integrate both sides**:
   - The left side involves \(\frac{1}{y^2}\), so integrate it as follows:
     \[
     \int \frac{dy}{y^2} = \int 5x \, dx
     \]
   - The integral of \(\frac{1}{y^2}\) is \(-\frac{1}{y}\), and the integral of \(5x\) is \(\frac{5x^2}{2}\).

   After integration, we get:
   \[
   -\frac{1}{y} = \frac{5x^2}{2} + C
   \]
   where \(C\) is the constant of integration.

3. **Solve for \(y\)**:
   Multiply both sides by \(-1\) to get:
   \[
   \frac{1}{y} = -\frac{5x^2}{2} - C
   \]

   Finally, take the reciprocal of both sides to solve for \(y\):
   \[
   y = \frac{1}{- \frac{5x^2}{2} - C}
   \]

Thus, the explicit general solution to the differential equation is:

\[
y = \frac{1}{- \frac{5x^2}{2} - C}
\]

---

Would you like more details on any of these steps or have any questions?

### 5 Related Questions:
1. How would the solution change if the right-hand side were different, e.g., \(y' = kxy^2\)?
2. What is the significance of the constant \(C\) in the solution?
3. Can this method be applied to non-separable differential equations?
4. How would we solve this if initial conditions were given, such as \(y(0) = 1\)?
5. What is the general approach to solving differential equations using separation of variables?

### Tip:
Always check if a differential equation is separable, as this method simplifies many first-order problems.

Find the explicit general solution to the following differential equation. A prime denotes a derivative with respect to x.

 y' = 5xy^2

Discover how to solve the differential equation y' = 5xy^2 by separation of variables. This step-by-step solution walks through separating variables and integrating both sides to arrive at the explicit general solution. Suitable for students learning Calculus I/II.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation