```python
import loader
from sympy import *
init_printing()
from root.solver import *
```

#### Find the general solution of  $y^{(4)} - 4y''' + 4y'' = 0$


```python
yc, p = nth_order_const_coeff(1, -4, 4, 0, 0)
p.display()
```


$\displaystyle \text{Characteristic equation: }$



$\displaystyle r^{4} - 4 r^{3} + 4 r^{2} = 0$



$\displaystyle \text{Roots: }\left\{ \begin{array}{ll}r_{1,2} = 0\\r_{3,4} = 2\\\end{array} \right.$



$\displaystyle \text{General Solution: }$



$\displaystyle y = C_{1} + C_{2} t + C_{3} e^{2 t} + C_{4} t e^{2 t}$


#### Find the general solution of  $y^{(6)} + y = 0$


```python
yc, p = nth_order_const_coeff(1, 0, 0, 0, 0, 0, 1)
p.display()
```


$\displaystyle \text{Characteristic equation: }$



$\displaystyle r^{6} + 1 = 0$



$\displaystyle \text{Roots: }\left\{ \begin{array}{ll}r_{1} = i\\r_{2} = - i\\r_{3} = - \frac{\sqrt{3}}{2} - \frac{i}{2}\\r_{4} = - \frac{\sqrt{3}}{2} + \frac{i}{2}\\r_{5} = \frac{\sqrt{3}}{2} - \frac{i}{2}\\r_{6} = \frac{\sqrt{3}}{2} + \frac{i}{2}\\\end{array} \right.$



$\displaystyle \text{General Solution: }$



$\displaystyle y = C_{1} \sin{\left(t \right)} + C_{2} \cos{\left(t \right)} + C_{3} e^{- \frac{\sqrt{3} t}{2}} \sin{\left(\frac{t}{2} \right)} + C_{4} e^{- \frac{\sqrt{3} t}{2}} \cos{\left(\frac{t}{2} \right)} + C_{5} e^{\frac{\sqrt{3} t}{2}} \sin{\left(\frac{t}{2} \right)} + C_{6} e^{\frac{\sqrt{3} t}{2}} \cos{\left(\frac{t}{2} \right)}$


3d6157fb16a89d8aa0e6f90946945568562dd7f4
ipynb
Jupyter Notebook
notebooks/higher-order-homogeneous-ode-constant-coefficients.ipynb
kaiyingshan/ode-solver
30c6798efe9c35a088b2c6043493470701641042
MIT
2019-02-17T23:15:20.000Z
2019-02-17T23:15:27.000Z
notebooks/higher-order-homogeneous-ode-constant-coefficients.ipynb
kaiyingshan/ode-solver
30c6798efe9c35a088b2c6043493470701641042
MIT
notebooks/higher-order-homogeneous-ode-constant-coefficients.ipynb
kaiyingshan/ode-solver
30c6798efe9c35a088b2c6043493470701641042
MIT
Qwen/Qwen-72B
1. YES
2. YES