---## 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

See similar textbooks

Related questions

Q: I) Consider the function: y = + C,e2x + C,e-5x + Czxe-5x

Q: Q.4. Solve: y" – y' + y = Vxe*

Q: Which of the following equations is non homogeneous differential equation? O a. dz = (z + y)? O b.…

Q: y" – 2y' + y = -3 –x+x, y(0) = -2, y(0) = 1.

Q: Rewrite the following differential equation using differential operators. 1. y" dºy Зу" + 4у d³y 12e…

A: The given differential equations are 1) y"'-3y"+4y= 12e2x+4e3x 2) y"""+4y"'= 2tanx

Q: lve the following partial differential equation by using Lagrange method: du = -(x² – y²) du

Q: Classify each of the equations above as autonomous, separable, linear, homogeneous, exact,…

A: I am solving questions ( B I and J ) as per the guidelines we can solve max 3 sub part of a question…

Q: d the values of az/ax and öz/ay at the point z - xy + yz + y' - 2 = 0, (1, 1, 1)

A: Given that z^3-xy+yz+y^3-2=0 To find the partial derivatives at point (1,1,1)

Q: Solve the nonhomogeneous equation? .3. x*y" + x²y" – 2xy' + 2y = x³ In x. -

Q: 6. Solve these equations y' (t) - ay(t) = Qet, starting from y(0) = 2: y'(t) − y(t) = 8e-3t y'(t) +…

Q: (x3 +) dx + (y + Inx) dy =0, is exact differential equation. Select one: O True O False Jump to...…

Q: 1.) Determine the order of the differential equation and classify it as linear or non-linear. y" +…

A: Linear /non linear differential equations

Q: dy y dx 2x = 3y2 – 2 when y = 1 and x = 1 in terms of x. %3D - -

Q: Find the general solution to the following equations. 1. x²y" — (y')² – 2xy' = 0

Q: Seleccione una: Oa. Cx у Ob. x?y-tanx +y?- c c-x? y= 2+ xe" Od. y=1+ Cr?+ 1)¬1/2

Q: Solve the given differential equation y liv) -81y = 0 %3D O a. y- c,e + cze 9* + C3cos3x + C48in3r +…

Q: Q1) Solve the following partial differential equation Ux + Uy = (cos x + sin y). U

A: 0

Q: Please reply as soon as posible. Consider the function: y = C1e2x + C2xe2x + C3e-5x Find a linear…

Q: The region bounded by f(x) 2x² + 12x + 14, x = = 0 is rotated about the y-axis. Find the volume of…

Q: QI) Solve the following partial differential equation Ux + Uy = (sec x + sin y). U

Q: Rocky Mountain Chocolate Factory (RMCF) founder and president Frank Crail employs 220 people in 361…

A: for net price equivalent rate determine first find the complement of the chain discount rate…

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 +

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Trending now

This is a popular solution!

Step by step

Solved in 4 steps with 4 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

SOLVE STEP BY STEP IN DIGITAL FORMAT 3. Is the series 2n convergent or divergent? Hint: quotient criterion
Graph the slope field for the equation dx/dy = -x(x2-4) and attach it to the solution. Draw (by hand or using technology) several integral curbves in different parts of the slope field. Find the equilibruim solutions and determine their stability.
H3. Solve the following differential equations dfy -2 dy +4 +4y = x-e-r, dx² dx dfy dy - -6 +9y=e³x (x²+7) dx2 dx 1) 2)
please send handwritten solution
2. The function Y1 x? is one solution of the differential equation x²y" – 2xy' + 2y = 0. Use | reduction of order to find a second solution y2. poir n of each differential equ below. [6 point
a-3 OR Find x' amd 'y' if YH
: O Reduce the DE. 4x²y + 3x(2y² + 2) = e* to linear equation then solve it. dx Solve the DE. y"+y' = 3 + 10sinx
dy = x - Y. da Consider the differential equation (A) At the point (1.5, – 1), the direction field has a slope of (B) At the point (– 1,1), the direction field has a slope of