Show that the second-order differential equation y″=F(x,y,y′) can be reduced to a system of two first-order differential equationsdydx=z,dzdx=F(x,y,z).Can something similar be done to the nth-order differential equationy(n)=F(x,y,y′,y″,…,y(n−1))?

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Let z=y′=dydx, thendzdx=z′=y″The equation y″=F(x,y,y′) can be written asdydx=zdzdx=F(x,y,z)Let z1=y′=dydx′, thendz1dx=z1′=y″Let z2=z1′=y″, thendz2dx=z2′=y‴In the same way as before zn−1=zn−2′=yn−1, thendzn−1dx=zn−1′=ynThis gives the following system of differential equations of first orderdydx=z1,dz1dx=z2,dz2dx=z3,…dzn−2dx=zn−1,dz1dx=F(x,y,z1,z2,…,zn−1)

Show that the second-order differential equation y″ = F(x, y, y′) can be reduced to a system of two first-order differential equations \frac{dy}{dx}=z

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