https://file.aiquickdraw.com/mathbot/file/1725589794349gt30505n.jpg

The uploaded image shows the following differential equation:

\[
6\left(2y + 5\right)(1 - y^2) = -2y - 6.
\]

Let's solve this differential equation. I'll begin by simplifying and rearranging it into a form suitable for solving.

### Step 1: Rearrange and Simplify

Given equation:
\[
6\left(2y + 5\right)(1 - y^2) = -2y - 6.
\]

I will simplify and solve it.After simplifying the given equation, it turns into:

\[
6y^3 + 15y^2 - 7y = 18.
\]

This cubic equation can be quite complex to solve exactly by hand. The solutions involve cubic roots and complex numbers. Here are the exact symbolic solutions:

1. \(-\frac{5}{6} + \left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right)\left(\frac{47}{108} + \frac{\sqrt{50483}i}{216}\right)^{1/3} + \frac{13}{12\left(-\frac{1}{2} - \frac{\sqrt{3}i}{2}\right)\left(\frac{47}{108} + \frac{\sqrt{50483}i}{216}\right)^{1/3}}\)
2. \(-\frac{5}{6} + \frac{13}{12\left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right)\left(\frac{47}{108} + \frac{\sqrt{50483}i}{216}\right)^{1/3}} + \left(-\frac{1}{2} + \frac{\sqrt{3}i}{2}\right)\left(\frac{47}{108} + \frac{\sqrt{50483}i}{216}\right)^{1/3}\)
3. \(-\frac{5}{6} + \frac{13}{12\left(\frac{47}{108} + \frac{\sqrt{50483}i}{216}\right)^{1/3}} + \left(\frac{47}{108} + \frac{\sqrt{50483}i}{216}\right)^{1/3}\)

These roots are quite complicated due to the nature of cubic equations.

Would you like a numerical approximation of these solutions, or do you need further assistance with the interpretation of these symbolic forms?

Solve the equation 6(2y + 5)(1 - y^2) = -2y - 6 using ODE.

Explore how to solve the cubic equation 6(2y + 5)(1 - y^2) = -2y - 6 step by step. This problem involves concepts from differential equations and algebra, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation