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Reduction of Order: Finding Second Solutions for Differential Equations, Study Guides, Projects, Research of Differential Equations

A step-by-step guide on how to use the Reduction of Order method to find a second solution for the given differential equation: x2y′′ −2y = 0, with y1 = x2. the calculations and discussions on the process.

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/27/2022

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Overview An Example Double Check Discussion
Reduction of Order
Bernd Schr¨
oder
Bernd Schr¨
oder Louisiana TechUniversity, College of Engineering and Science
Reduction of Order
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Reduction of Order

Bernd Schr¨oder

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

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What is Reduction of Order?

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

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What is Reduction of Order?

  1. Typically, reduction of order is applied to second order linear differential equations of the form y′′^ + P(x)y′^ + Q(x)y = 0.
  2. We must already have one solution y 1 of the equation.

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

What is Reduction of Order?

  1. Typically, reduction of order is applied to second order linear differential equations of the form y′′^ + P(x)y′^ + Q(x)y = 0.
  2. We must already have one solution y 1 of the equation.
  3. Reduction of order assumes there is a second, linearly independent solution of a the form y = uy 1.

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

What is Reduction of Order?

  1. Typically, reduction of order is applied to second order linear differential equations of the form y′′^ + P(x)y′^ + Q(x)y = 0.
  2. We must already have one solution y 1 of the equation.
  3. Reduction of order assumes there is a second, linearly independent solution of a the form y = uy 1.
  4. There are two ways to proceed. 4.1 We can substitute y = uy 1 into the equation. This leads to a first order differential equation, which explains the name.

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

What is Reduction of Order?

  1. Typically, reduction of order is applied to second order linear differential equations of the form y′′^ + P(x)y′^ + Q(x)y = 0.
  2. We must already have one solution y 1 of the equation.
  3. Reduction of order assumes there is a second, linearly independent solution of a the form y = uy 1.
  4. There are two ways to proceed. 4.1 We can substitute y = uy 1 into the equation. This leads to a first order differential equation, which explains the name. 4.2 Or, we can use the formula y 2 = y 1

∫ (^) e− ∫^ P dx y^21 dx,^ which is obtained by doing the above substitution symbolically.

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

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Use Reduction of Order to Find a Second

Solution for x^2 y′′^ − 2 y = 0, given that y 1 = x^2.

y = ux^2

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Use Reduction of Order to Find a Second

Solution for x^2 y′′^ − 2 y = 0, given that y 1 = x^2.

y = ux^2 y′^ = u′x^2 + u 2 x

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Use Reduction of Order to Find a Second

Solution for x^2 y′′^ − 2 y = 0, given that y 1 = x^2.

y = ux^2 y′^ = u′x^2 + u 2 x y′′^ = u′′x^2 + u′ 4 x + u 2 x^2

u′′x^2 + u′ 4 x + u 2

− 2 ux^2 = 0

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Use Reduction of Order to Find a Second

Solution for x^2 y′′^ − 2 y = 0, given that y 1 = x^2.

y = ux^2 y′^ = u′x^2 + u 2 x y′′^ = u′′x^2 + u′ 4 x + u 2 x^2

u′′x^2 + u′ 4 x + u 2

− 2 ux^2 = 0 u′′x^4 + u′ 4 x^3 + u 2 x^2 − 2 ux^2 = 0

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Use Reduction of Order to Find a Second

Solution for x^2 y′′^ − 2 y = 0, given that y 1 = x^2.

y = ux^2 y′^ = u′x^2 + u 2 x y′′^ = u′′x^2 + u′ 4 x + u 2 x^2

u′′x^2 + u′ 4 x + u 2

− 2 ux^2 = 0 u′′x^4 + u′ 4 x^3 + u 2 x^2 − 2 ux^2 = 0 u′′x^4 + u′ 4 x^3 = 0 u′′x + u′ 4 = 0

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Use Reduction of Order to Find a Second

Solution for x^2 y′′^ − 2 y = 0, given that y 1 = x^2.

y = ux^2 y′^ = u′x^2 + u 2 x y′′^ = u′′x^2 + u′ 4 x + u 2 x^2

u′′x^2 + u′ 4 x + u 2

− 2 ux^2 = 0 u′′x^4 + u′ 4 x^3 + u 2 x^2 − 2 ux^2 = 0 u′′x^4 + u′ 4 x^3 = 0 u′′x + u′ 4 = 0 v := u′

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

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Use Reduction of Order to Find a Second

Solution for x^2 y′′^ − 2 y = 0, given that y 1 = x^2.

dv dx =^ −

4 v x

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science

logo

Use Reduction of Order to Find a Second

Solution for x^2 y′′^ − 2 y = 0, given that y 1 = x^2.

dv dx =^ −

4 v x dv v =^ −^4

dx x

Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science