How to express a 3rd order differential equation into three first order differential equations?
Explain.
Let's 3rd order differential equation is y′′′(t)=f(t,y,y′,y′′)y'''(t) = f(t, y, y', y'')y′′′(t)=f(t,y,y′,y′′). let introduce new vector-function x→(t)\overrightarrow{x}(t)x(t)
x→=[x1(t)x2(t)x3(t)]=[y(t)y′(t)y′′(t)]\overrightarrow{x} = \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} = \begin{bmatrix} y(t) \\ y'(t) \\ y''(t) \end{bmatrix}x=⎣⎡x1(t)x2(t)x3(t)⎦⎤=⎣⎡y(t)y′(t)y′′(t)⎦⎤
then
dx→dt=ddt[y(t)y′(t)y′′(t)]=[x2(t)x3(t)f(t,x1(t),x2(t),x3(t)]\frac{d \overrightarrow{x}}{d t} = \frac{d }{d t} \begin{bmatrix} y(t) \\ y'(t) \\ y''(t) \end{bmatrix} = \begin{bmatrix} x_2(t) \\ x_3(t) \\ f(t, x_1(t), x_2(t), x_3(t) \end{bmatrix}dtdx=dtd⎣⎡y(t)y′(t)y′′(t)⎦⎤=⎣⎡x2(t)x3(t)f(t,x1(t),x2(t),x3(t)⎦⎤
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