**Problem Statement:**Apply the method of reduction of order (not its formula) to find the second solution for the differential equation:\[ 6y'' + y' - y = 0 \]given that one solution is \[ y_1 = e^{\frac{x}{3}} \] **Explanation:**The task is to find a second linearly independent solution to the given second-order homogeneous linear differential equation using the method of reduction of order. This method involves assuming a solution of a specific form and using known solutions to reduce the order of the differential equation, making it easier to solve.**Steps to Solve:**1. Start with the known solution $ y_1 = e^{\frac{x}{3}} $.2. To find a second solution $ y_2 $, assume $ y_2 = v(x) y_1 $, where $ v(x) $ is a function to be determined.3. Substitute $ y_2 $ into the original differential equation to derive an equation for $ v(x) $.4. Solve the resulting first-order differential equation to find the function $ v(x) $.5. Multiply $ v(x) $ by $ y_1 $ to find the second solution $ y_2 $.The process typically involves integration and algebraic manipulation, usually resulting in an expression that provides the general solution to the differential equation.

Write the given differential equation 6y''+y'-y=0 in standard form by dividing by 6 throughout.…

Answered: Apply the method of reduction of order…

Math

Advanced Math

Apply the method of reduction of order (not itť's formula) to find the second solution if 6y"+ y'-y = 0 y, = e3

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

**Problem Statement:**

Apply the method of reduction of order (not its formula) to find the second solution for the differential equation:

\[ 6y'' + y' - y = 0 \]

given that one solution is

\[ y_1 = e^{\frac{x}{3}} \]

**Explanation:**

The task is to find a second linearly independent solution to the given second-order homogeneous linear differential equation using the method of reduction of order. This method involves assuming a solution of a specific form and using known solutions to reduce the order of the differential equation, making it easier to solve.

**Steps to Solve:**

1. Start with the known solution $ y_1 = e^{\frac{x}{3}} $.

2. To find a second solution $ y_2 $, assume $ y_2 = v(x) y_1 $, where $ v(x) $ is a function to be determined.

3. Substitute $ y_2 $ into the original differential equation to derive an equation for $ v(x) $.

4. Solve the resulting first-order differential equation to find the function $ v(x) $.

5. Multiply $ v(x) $ by $ y_1 $ to find the second solution $ y_2 $.

The process typically involves integration and algebraic manipulation, usually resulting in an expression that provides the general solution to the differential equation.

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