### Differential Equation with Initial Condition#### Problem Statement:Given the differential equation:\[ xy^2 \frac{dy}{dx} = y^3 - x^3 \]with the initial condition:\[ y(1) = 2 \]#### Explanation:This is a first-order, non-linear ordinary differential equation. The goal is to find the function \( y(x) \) that satisfies both the differential equation and the initial condition provided.### Steps to Solve:1. **Substitute the Initial Condition:** Start by verifying the initial condition. When \( x = 1 \), \( y(1) = 2 \).2. **Separation of Variables (if applicable):** Try to separate the variables \( x \) and \( y \) to integrate each side independently.3. **Integration:** After separating the variables, integrate both sides to find the general solution of the differential equation.4. **Apply Initial Condition:** Use the initial condition \( y(1) = 2 \) to find the specific constant and obtain the particular solution.This process will help determine the explicit form of the function \( y(x) \).### Note:Further steps, including specific techniques and integration methods, may be necessary depending on the behavior of the given differential equation. Please consult additional resources or educational material on solving first-order non-linear differential equations for detailed methodologies.### Visual Aid:- **No additional graphs or diagrams are presented in the image.** However, plotting the function \( y(x) \) after solving the equation can provide visual insights into the behavior of the solution.### Additional Resources:- For more examples and detailed explanations, you can refer to calculus or differential equations textbooks.- Online educational platforms may offer video tutorials and practice problems for similar differential equations.

To solve the differential equation of the form, dydx=f(x,y)g(x,y) where f and g are homogeneous…

Answered: 9. xy = y' -x', y(1)=2 dx

9. xy = y' -x', y(1)=2 dx

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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