---## Example Problem: Solving a Linear Differential EquationConsider the linear differential equation:\[ x^3 y'' - xy' + y = 0 \]### Steps to Solve:**(a)** Verify that $ y_1 = x $ is a solution to this differential equation (i.e., show that $ y_1 = x $ satisfies the DE).**(b)** Use the method of reduction of order to find a second linearly independent solution $ y_2 $.**Hint:** For reduction of order, suppose that $ y_2 = uy_1 $.---### Solution Outline:1. **Verification of $ y_1 = x $ as a Solution:** - To verify that $ y_1 = x $ is a solution, substitute $ y_1 = x $ into the differential equation and confirm that the left-hand side equals zero.2. **Reduction of Order:** - Given the first solution $ y_1 = x $, find the second solution by assuming $ y_2 = uy_1 $, where $ u $ is a function of $ x $. - Derive the necessary expressions and substitute back into the original differential equation to solve for $ u $.---### Detailed Solution Process:**Verification:**1. Let $ y_1 = x $2. Compute the required derivatives: - $ y_1' = 1 $ - $ y_1'' = 0 $3. Substitute these into the differential equation $ x^3 y'' - xy' + y $: \[ x^3 (0) - x(1) + x = 0 \]4. Simplify to show: \[ -x + x = 0 \] This confirms that $ y_1 = x $ satisfies the differential equation.**Reduction of Order:**1. Assume $ y_2 = uy_1 = ux $2. Compute the derivatives needed for substitution: - $ y_2' = u'x + u $ - $ y_2'' = u''x + 2u' $3. Substitute $ y_2, y_2', y_2'' $ into the differential equation: \[ x^3 (u''x + 2u') - x(u'x +

We are given the linear differential equation x3y'' - xy' + y = 0. For (a), We need to first…

Consider the linear differential equation ay" - xy' + y = 0. (a) Verify that y = x is a solution to this DE. (i.e., show that y1 = I satisfies the DE.) (b) Use the method of reduction of order to find a second lin. indep. solution y2. Hint: For reduction of order, suppose that y2 = uY1.

Consider the linear differential equation ay" - xy' + y = 0. (a) Verify that y = x is a solution to this DE. (i.e., show that y1 = I satisfies the DE.) (b) Use the method of reduction of order to find a second lin. indep. solution y2. Hint: For reduction of order, suppose that y2 = uY1.

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

---

## Example Problem: Solving a Linear Differential Equation

Consider the linear differential equation:
\[ x^3 y'' - xy' + y = 0 \]

### Steps to Solve:

**(a)** Verify that $ y_1 = x $ is a solution to this differential equation (i.e., show that $ y_1 = x $ satisfies the DE).

**(b)** Use the method of reduction of order to find a second linearly independent solution $ y_2 $.

**Hint:** For reduction of order, suppose that $ y_2 = uy_1 $.

---

### Solution Outline:

1. **Verification of $ y_1 = x $ as a Solution:**
- To verify that $ y_1 = x $ is a solution, substitute $ y_1 = x $ into the differential equation and confirm that the left-hand side equals zero.

2. **Reduction of Order:**
- Given the first solution $ y_1 = x $, find the second solution by assuming $ y_2 = uy_1 $, where $ u $ is a function of $ x $.
- Derive the necessary expressions and substitute back into the original differential equation to solve for $ u $.

---

### Detailed Solution Process:

**Verification:**

1. Let $ y_1 = x $
2. Compute the required derivatives:
- $ y_1' = 1 $
- $ y_1'' = 0 $
3. Substitute these into the differential equation $ x^3 y'' - xy' + y $:
\[
x^3 (0) - x(1) + x = 0
\]
4. Simplify to show:
\[
-x + x = 0
\]
This confirms that $ y_1 = x $ satisfies the differential equation.

**Reduction of Order:**

1. Assume $ y_2 = uy_1 = ux $
2. Compute the derivatives needed for substitution:
- $ y_2' = u'x + u $
- $ y_2'' = u''x + 2u' $
3. Substitute $ y_2, y_2', y_2'' $ into the differential equation:
\[ x^3 (u''x + 2u') - x(u'x +

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