solve the given initial-value problem ### Differential Equation Solution Example#### Problem Statement:Solve the following differential equation with the given initial value:\[x y^2 \frac{dy}{dx} = y^3 - x^3, \quad y(1) = 2\]### Explanation:1. **Understanding the equation:** - The equation is given in the form $ x y^2 \frac{dy}{dx} = y^3 - x^3 $. - This is a first-order, non-linear differential equation.2. **Initial Condition:** - The initial condition provided is $ y(1) = 2 $. 3. **Procedure:** - To solve this, one might separate variables or use substitution methods, depending on what's most appropriate for the form of the differential equation.### Detailed Steps:1. Rewrite the differential equation: \[ x y^2 \frac{dy}{dx} = y^3 - x^3 \] Dividing both sides by $ y^2 $, we get: \[ x \frac{dy}{dx} = y - \frac{x^3}{y^2} \]2. Multiplying both sides by $ dx $ and separating variables, we obtain: \[ \frac{y^2}{y^3 - x^3} dy = \frac{1}{x} dx \]3. Integrate both sides within suitable limits.### Conclusion:- After integration, apply the initial condition $ y(1) = 2 $ to find the constant of integration.- Solve the resulting equation to get the explicit form of $ y $ in terms of $ x $.Please check an appropriate reference or differential equations resource for the integration steps and final solution. This equation might require more advanced integration techniques or numerical methods for a full solution.

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Answered: 2. 3. メーX, y0-2 | dx

Math

Advanced Math

2. 3. メーX, y0-2 | dx

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

solve the given initial-value problem

### Differential Equation Solution Example

#### Problem Statement:
Solve the following differential equation with the given initial value:

\[
x y^2 \frac{dy}{dx} = y^3 - x^3, \quad y(1) = 2
\]

### Explanation:
1. **Understanding the equation:**
- The equation is given in the form $ x y^2 \frac{dy}{dx} = y^3 - x^3 $.
- This is a first-order, non-linear differential equation.

2. **Initial Condition:**
- The initial condition provided is $ y(1) = 2 $.

3. **Procedure:**
- To solve this, one might separate variables or use substitution methods, depending on what's most appropriate for the form of the differential equation.

### Detailed Steps:
1. Rewrite the differential equation:
\[
x y^2 \frac{dy}{dx} = y^3 - x^3
\]
Dividing both sides by $ y^2 $, we get:
\[
x \frac{dy}{dx} = y - \frac{x^3}{y^2}
\]

2. Multiplying both sides by $ dx $ and separating variables, we obtain:
\[
\frac{y^2}{y^3 - x^3} dy = \frac{1}{x} dx
\]

3. Integrate both sides within suitable limits.

### Conclusion:
- After integration, apply the initial condition $ y(1) = 2 $ to find the constant of integration.
- Solve the resulting equation to get the explicit form of $ y $ in terms of $ x $.

Please check an appropriate reference or differential equations resource for the integration steps and final solution. This equation might require more advanced integration techniques or numerical methods for a full solution.

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