Find an explicit solution of the given initial-value problem. ### Differential Equation Example for Educational PurposesConsider the following differential equation and initial condition:\[ x^2 \frac{dy}{dx} = y - xy \]with the initial condition:\[ y(-1) = -1\]### Explanation:This is a first-order differential equation involving the variables $x$ and $y$, and their derivatives. The equation specifies how the function $y(x)$ changes with respect to $x$.**Steps to Solve:**1. **Separate Variables:** To solve this differential equation, we can try to separate variables $x$ and $y$. 2. **Integrate Both Sides:** Integrate both sides with respect to their respective variables.3. **Apply the Initial Condition:** Use the initial condition $y(-1) = -1$ to solve for any constant of integration that arises.### Graphical Interpretation:While the current content does not include a graph or diagram, it is helpful to visualize the solution. Typically, the solution of such a differential equation could be plotted as a curve in the $xy$-plane, representing how $y$ changes with $x$.### Usefulness in Education:Understanding and solving differential equations is fundamental in many fields such as physics, engineering, economics, and several other sciences. This exercise helps illustrate important techniques and concepts such as separation of variables, integration, and initial value problems.

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Answered: dy 2 %3D

Math

Advanced Math

dy 2 %3D

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

Find an explicit solution of the given initial-value
problem.

### Differential Equation Example for Educational Purposes

Consider the following differential equation and initial condition:

\[
x^2 \frac{dy}{dx} = y - xy
\]

with the initial condition:

\[
y(-1) = -1
\]

### Explanation:

This is a first-order differential equation involving the variables $x$ and $y$, and their derivatives. The equation specifies how the function $y(x)$ changes with respect to $x$.

**Steps to Solve:**

1. **Separate Variables:** To solve this differential equation, we can try to separate variables $x$ and $y$.

2. **Integrate Both Sides:** Integrate both sides with respect to their respective variables.

3. **Apply the Initial Condition:** Use the initial condition $y(-1) = -1$ to solve for any constant of integration that arises.

### Graphical Interpretation:

While the current content does not include a graph or diagram, it is helpful to visualize the solution. Typically, the solution of such a differential equation could be plotted as a curve in the $xy$-plane, representing how $y$ changes with $x$.

### Usefulness in Education:

Understanding and solving differential equations is fundamental in many fields such as physics, engineering, economics, and several other sciences. This exercise helps illustrate important techniques and concepts such as separation of variables, integration, and initial value problems.

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