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Euler Method (2nd-Order ODE)

TOP/Numerical Analysis/Ordinary differential equations
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Find the solution y=f(x) of the second-order ordinary differential equation y’’=F(x,y,y’) using the Euler method. Starting from y0=f(x0), y’0=f’(x0), calculate the solution for x0≦x≦xn.
\(\normalsize y''=F(x,y,y')\hspace{30px} y_0=f(x_0),\ y'_0=f'(x_0) \rightarrow\ y=f(x)\\\)
F(x,y,p(=y'))
x0
Initial conditions
y0
=f(x0) Initial conditions
y'0=p0
=f'(x0) Initial conditions
xn
x0≦x≦xn
[ n]Number of segments
\(\normalsize Euler's\ method\\ (1)\ y''=F(x,y,y'),\hspace{30px} y_0=f(x_0),\ y'_0=f'(x_0)\rightarrow\ y=f(x)\\ (2)\ y_{n+1}=y_n + h \cdot p_n +{\small O}(h^2), \\ \hspace{23px} p_{n+1}= p_n + h F(x_n,y_n,p_n) +{\small O}(h^2), \\ \hspace{23px} x_n=x_0+nh\\ \)

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Euler Method (2nd-Order ODE) | Keisan