Simple Linear 1

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Equations:

$$\begin{align}\dot{x}_{0}&=x_{1}\\\dot{x}_{1}&=9.0 \cdot x_{0}\end{align}$$

Description:

A simple second order linear equation with constant coefficients.

Keywords:

simple; linear; constant coefficients

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Plots:

C5 WPD

Initial value problems:

18 (PILANU)
19 (PILAU)

IVP 18

Class: PILANU

Initial conditions:

$$\begin{align}x_{0}(0)&=1.0\\x_{1}(0)&=1.0\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&=0.666666666666667 \cdot e^{3.0 \cdot t} + 0.333333333333333 \cdot e^{- 3.0 \cdot t}\\x_{1}(t)&=2.0 \cdot e^{3.0 \cdot t} - 1.0 \cdot e^{- 3.0 \cdot t}\end{align}$$

View test data:

Solver
Settings
Results

VNODE-LP

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.00500
c5: t_us=0.00390 , e_us=4.902744876744691e-13
c6: t_bd=10.000000 , e_bd=3.656250000000000e+00

VNODE-LP

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.00890
c5: t_us=0.00660 , e_us=1.136868377216160e-13
c6: t_bd=10.000000 , e_bd=1.164062500000000e+00

VNODE-LP

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.01290
c5: t_us=0.00860 , e_us=1.207922650792170e-13
c6: t_bd=10.000000 , e_bd=2.667968750000000e+00

VNODE-LP

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.05510
c5: t_us=0.04960 , e_us=4.737898962048348e-11
c6: t_bd=10.000000 , e_bd=2.613554687500000e+02

Generate graphics:

VNODE-LP

WPD for Problem 18

IVP 19

Class: PILAU

Initial conditions:

$$\begin{align}x_{0}(0)&=1.0\\x_{1}(0)&\subset~\left[ 0.5, 1.0\right]\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&\subset~e^{3.0 \cdot t} \cdot \left[ 0.583333333333333, 0.666666666666667\right] + e^{- 3.0 \cdot t} \cdot \left[ 0.333333333333333, 0.416666666666667\right]\\x_{1}(t)&\subset~e^{3.0 \cdot t} \cdot \left[ 1.75, 2.0\right] + e^{- 3.0 \cdot t} \cdot \left[ -1.25, -1.0\right]\end{align}$$