\[\begin{align}
f_2 \cdot \ddot{X(t)}+f_1 \cdot \dot{X(t)} +f_0 \cdot {X(t)} =F(t)
\end{align}
\]

\[\begin{align}
f_1 \cdot \dot{X(t)} +f_0 \cdot {X(t)} =F(t)
\end{align}
\]

\[\begin{align}

\dot{X(t)} = \frac {F(t)-f_0 \cdot {X(t)}}{f_1 }\\
\frac {X_{n+1}-X_n}{dt} = \frac {F(t)-f_0 \cdot {X(t)}}{f_1 }\\
X_{n+1}=X_n+dt\cdot\frac {F(t)-f_0 \cdot {X_{n}}}{f_1 }=X_n+dt\cdot f(t,X_n)

\end{align}
\]

\[\begin{align}
\begin{cases}

X_{n+1}=X_{n}+\frac{dt}{6}\left (k_1+2k_2+2k_3+k_4 \right) \\
k_1=f\left (t,X_n \right) \\
k_2=f\left (t+\frac {dt}{2},X_n+\frac {dt}{2}\cdot k_1 \right) \\
k_3=f\left (t+\frac {dt}{2},X_n+\frac {dt}{2}\cdot k_2 \right) \\
k_4=f\left (t+{dt},X_n+dt \cdot {k_3} \right)\\

\end{cases}

\end{align}
\]

\[\begin{align}
f_2 \cdot \ddot{X(t)}+f_1 \cdot \dot{X(t)} +f_0 \cdot {X(t)} =F(t)
\end{align}
\]

\[\begin{align}
\begin{cases}

a(t)=X(t) \\
b(t)= \dot a(t)=\dot X(t)

\end{cases}

\end{align}
\]

\[\begin{align}
\begin{cases}

f_2\cdot \dot b(t)+f_1\cdot b(t) +f_0\cdot a(t)= F(t) \\
b(t)=\dot a(t)

\end{cases}

\end{align}
\]

\[\begin{align}
\begin{cases}
\dot a(t)=b(t)\\
\dot b(t)= \frac {F(t)-(f_1\cdot b(t) +f_0\cdot a(t))}{f_2} \\
\end{cases}
\end{align}
\]

\[\begin{align}
\begin{cases}
\dot a_{n+1}=a_{n}+dt\cdot b(t) = a_{n}+dt\cdot f(t,a_n,b_n) \\

\dot b_{n+1}=b_{n}+dt\cdot \frac {F(t)-(f_1\cdot b(t) +f_0\cdot a(t))}{f_2} = b_{n}+dt\cdot g(t,a_n,b_n) \\
\end{cases}
\end{align}
\]

\[\begin{align}
\begin{cases}
\dot a_{n+1}=a_{n}+dt\cdot f(t,a_n,b_n) \\

\dot b_{n+1} = b_{n}+dt\cdot g(t,a_n,b_n) \\
\end{cases}

\end{align}
\]

\[\begin{bmatrix}
a_{n+1}\\b_{n+1}
\end{bmatrix}=
\begin{bmatrix}
a_n+\frac{dt}{6}\cdot (k_{1a}+2k_{2a}+2k_{3a}+k_{4a})\\
b_n+\frac{dt}{6}\cdot (k_{1b}+2k_{2b}+2k_{3b}+k_{4b})
\end{bmatrix}\\
\]

\[\begin{align}
\begin{cases}
\begin{bmatrix}
k_{1a}\\k_{1b}
\end{bmatrix}=
\begin{bmatrix}
f(t,a_n,b_n)\\
g(t,a_n,b_n)
\end{bmatrix}\\

\begin{bmatrix}
k_{2a}\\k_{2b}
\end{bmatrix}=
\begin{bmatrix}
f(t+\frac{dt}{2},a_n+\frac{dt}{2} \cdot k_{1a},b_n+\frac{dt}{2} \cdot k_{1b})\\
g(t+\frac{dt}{2},a_n+\frac{dt}{2} \cdot k_{1a},b_n+\frac{dt}{2} \cdot k_{1b})
\end{bmatrix}\\

\begin{bmatrix}
k_{3a}\\k_{3b}
\end{bmatrix}=
\begin{bmatrix}
f(t+\frac{dt}{2},a_n+\frac{dt}{2} \cdot k_{2a},b_n+\frac{dt}{2} \cdot k_{2b})\\
g(t+\frac{dt}{2},a_n+\frac{dt}{2} \cdot k_{2a},b_n+\frac{dt}{2} \cdot k_{2b})
\end{bmatrix}\\

\begin{bmatrix}
k_{4a}\\k_{4b}
\end{bmatrix}=
\begin{bmatrix}
f(t+dt,a_n+dt \cdot k_{3a},b_n+dt \cdot k_{3b})\\
g(t+dt,a_n+dt \cdot k_{3a},b_n+dt \cdot k_{3b})
\end{bmatrix}\\

\end{cases}

\end{align}
\]

\[\begin{align}
1 \cdot \ddot{X(t)}+0 \cdot \dot{X(t)} +0 \cdot {X(t)} =cos(t)\\
\dot{X(0)}=0\\
{X(0)} =0
\end{align}
\]

\[\begin{align}
{X(t)} =-cos(t)+1
\end{align}
\]

#include"RK_ODE.h"

#include<iostream>

using namespace std;

#define M 1

MatrixXd F2(double t) {

	MatrixXd res = MatrixXd::Identity(M, M);

	return res;

}

MatrixXd F1(double t) {

	MatrixXd res = MatrixXd::Zero(M, M);

	return res;

}

MatrixXd F0(double t) {

	MatrixXd res = MatrixXd::Zero(M, M);

	return res;

}

MatrixXd F(double t) {

	MatrixXd res = MatrixXd::Ones(M, 1);

	res(0, 0) = cos(t);

	return res;

}

int main() {

	RK_ODE ode(M, 0.1);

	ode.X.fill(0.);

	ode.dX.fill(0.);

	for (int i = 0; i < 1000; i++) {

		ode.Solve(F2, F1, F0, F);

		ode.WriteMotionAndForce("results1.txt", 0);

	}

	//ode.SaveSnapshoot("snapshoot.json");

	cout << ode._Mode << endl;

}