### Differential Equations#### Problem Description:Solve the given differential equation by using an appropriate substitution. The DE is of the form \[ \frac{dy}{dx} = f(Ax + By + C), \]which is given in equation (5) of Section 2.5.\[ \frac{dy}{dx} = \tan^2(x + y) \]This equation can be approached by making a proper substitution to transform it into a more solvable form. Refer to Section 2.5 for the steps on solving such differential equations.#### Steps for Solution:1. **Identify the substitution needed**: Determine an appropriate substitution where $ u = Ax + By + C $.2. **Transform and solve**: Rewrite the differential equation in terms of $ u $ and solve the resulting equation.3. **Revert to original variables**: After solving for $ u $, revert back to the original variables $ x $ and $ y $ to find the solution to the given differential equation.For detailed steps and explanations, please refer to the corresponding section in your textbook or learning material.#### Visual Explanation:- **Graphs/Diagrams**: If any specific graph or diagram is used, please refer to it in Section 2.5 for a better understanding of the substitution method.Feel free to reach out if you have any questions or need further clarifications on solving such differential equations.

We solve the given differential equation by the substitution of x+y=u and then proceed further as in…

Answered: Solve the given differential equation…

Solve the given differential equation by using an appropriate substitution. The DE is of the form = f(Ax + By + C), which is given in (5) of Section 2.5. dx dy = tan2(x + y) xp

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

### Differential Equations

#### Problem Description:
Solve the given differential equation by using an appropriate substitution. The DE is of the form

\[ \frac{dy}{dx} = f(Ax + By + C), \]

which is given in equation (5) of Section 2.5.

\[ \frac{dy}{dx} = \tan^2(x + y) \]

This equation can be approached by making a proper substitution to transform it into a more solvable form. Refer to Section 2.5 for the steps on solving such differential equations.

#### Steps for Solution:
1. **Identify the substitution needed**: Determine an appropriate substitution where $ u = Ax + By + C $.
2. **Transform and solve**: Rewrite the differential equation in terms of $ u $ and solve the resulting equation.
3. **Revert to original variables**: After solving for $ u $, revert back to the original variables $ x $ and $ y $ to find the solution to the given differential equation.

For detailed steps and explanations, please refer to the corresponding section in your textbook or learning material.

#### Visual Explanation:
- **Graphs/Diagrams**: If any specific graph or diagram is used, please refer to it in Section 2.5 for a better understanding of the substitution method.

Feel free to reach out if you have any questions or need further clarifications on solving such differential equations.

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