Homogeneous EquationsExpected Educational ResultsConstant Coefficients Method for IVPsInvestigation 06Use Technology to Solve Characteristic EquationUse Technology to Solve for Constants in

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 4.2**– Homogeneous Linear Equations: The General Solution

**Objective 10–1**: I can identify if two or more functions are linearly-independent.**Objective 10–2**: I can identify the characteristic equation for -degree homogeneous linear ODEs.${n}^{\text{th}}$ **Objective 10–3**: I can find the most general solution to -degree homogeneous linear ODEs.${n}^{\text{th}}$

Identify the characteristic equation for the homogeneous linear DE.

Completely solve the characteristic equation.

Use the roots of the characteristic equation for the homogeneous,

, solution.${y}_{h}(t)$ Use system of equations to solve for the constants.

Solve the following

;${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime}+4{y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}+3y=0$ and$y(0)=0$ ${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(0)=20$ ;${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime}+6{y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}+9y=0$ and$y(0)=1$ ${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(0)=2$ ,${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime \prime}(t)+{y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime}(t)-6{y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(t)=0$ ,$y(0)=2$ ,${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(0)=-3$ ${y}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime \prime}(0)=1$

**Example 01**: Solve

**Solution**:

The characteristic equation is

`1``(* Example: Solve y'''(x) - 3y''(x) - 40 y'(x) = 0 *)`

2`(* Solve characteristic equation *)`

3`Solve[r^3 - 3r^2 - 40r == 0, r]`

**Warnings**:

Be very

**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Mathematica*syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Thus,

Solve for the constants in the IVP:

Solve the following system of equations for

`x1``(* Example: Solve y'''(x) - 3y''(x) - 40 y'(x) = 0 *)`

2`(* Solve for constants *)`

3`(* Dolve has 2 arguments *)`

4`(* 1. all equations with grouped with braces *)`

5`(* 2. all constants grouped with braces *)`

6```
```

7`Solve[{c1 + c2 + c3 == 0, -5c1 + 8c3 == -2, 25c1 + 64c3 == 1}, {c1, c2, c3}]`

**Warnings**:

Be very

**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Mathematica*syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

`xxxxxxxxxx`

91`(* Example: Solve y'''(x) - 3y''(x) - 40 y'(x) = 0 *)`

2`(* where y[0]==0, y'[0]==-2, y''[0]==1 *)`

3```
```

4`(* DSolve uses function notation for dependent variable; and has 3 arguments *)`

5`(* 1. ODE equation with == for =; grouped with initial conditions *)`

6`(* 2. dependent variable, in function notation *)`

7`(* 3. independent variable *)`

8```
```

9`DSolve[{y'''[x] - 3y''[x] - 40 y'[x] == 0, y[0]==0, y'[0]==-2, y''[0]==1}, y[x], x]`

**Warnings**:

Be very

**careful**with the*syntax*.*Syntax*is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic*Mathematica*syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

**Last Modified**: Monday, 6 September 2020 17:33 EDT