Transformation 2 How to rewrite a system that are not first-order as first- order system? 4口4①，1无2)只0 Linear of Differential Equations
Transformation 2 How to rewrite a system that are not first-order as firstorder system? Suppose we have a system for two unknown functions x1 and x2 that is of the form F(t, x1, x 0 1 ,··· , x (m) 1 , x2, x 0 2 ,··· , x (n) 2 ) = 0, G(t, x1, x 0 1 ,··· , x (m) 1 , x2, x 0 2 ,··· , x (n) 2 ) = 0, here m and n are the orders of the highest order derivatives of x1 and x2 that appear in the two equations. Linear Systems of Differential Equations
Transformation 2 How to rewrite a system that are not first-order as first- order system? Suppose we have a system for two unknown functions x1 and x2 that is of the form F,1,,2,…,) =0, G,,,…,m,2,,,) =0, 4口①，1，元卡2000 Linear of Differential Equations
Transformation 2 How to rewrite a system that are not first-order as firstorder system? Suppose we have a system for two unknown functions x1 and x2 that is of the form F(t, x1, x 0 1 ,··· , x (m) 1 , x2, x 0 2 ,··· , x (n) 2 ) = 0, G(t, x1, x 0 1 ,··· , x (m) 1 , x2, x 0 2 ,··· , x (n) 2 ) = 0, here m and n are the orders of the highest order derivatives of x1 and x2 that appear in the two equations. Linear Systems of Differential Equations
Transformation 2 How to rewrite a system that are not first-order as first- order system? Suppose we have a system for two unknown functions x1 and x2 that is of the form ）=0, G,,,…,xm）,2,…,0) .(m) =0, here m and n are the orders of the highest order derivatives of xI and x2 that appear in the two equations. 4口0,4元3000 Linear of Differential Equations
Transformation 2 How to rewrite a system that are not first-order as firstorder system? Suppose we have a system for two unknown functions x1 and x2 that is of the form F(t, x1, x 0 1 ,··· , x (m) 1 , x2, x 0 2 ,··· , x (n) 2 ) = 0, G(t, x1, x 0 1 ,··· , x (m) 1 , x2, x 0 2 ,··· , x (n) 2 ) = 0, here m and n are the orders of the highest order derivatives of x1 and x2 that appear in the two equations. Linear Systems of Differential Equations
Suppose that it is possible to solve algebraically for (n) and x2, 4口4①，1无2)只0 Linear of Differential Equations
Suppose that it is possible to solve algebraically for x (m) 1 and x (n) 2 , so that we can replace the system by the equivalent system x (m) 1 = f(t, x1, x 0 1 ,··· , x (m−1) 1 , x2, x 0 2 ,··· , x (n−1) 2 ), x (n) 2 = g(t, x1, x 0 1 ,··· , x (m−1) 1 , x2, x 0 2 ,··· , x (n−1) 2 ). Set u1 = x1, u2 = x 0 1 , u3 = x 00 1 ,··· , um = x (m−1) 1 , um+1 = x2, um+2 = x 0 2 , um+3 = x 00 2 ,··· , um+n = x (n−1) 2 . Linear Systems of Differential Equations
Suppose that it is possible to solve algebraically for andso that we can replace the system by the equivalent system x0=f6,,…,m-,2…0-。 四=8,1，…m-,2,,…- 4口4①，1无2)只0 Linear of Differential Equations
Suppose that it is possible to solve algebraically for x (m) 1 and x (n) 2 , so that we can replace the system by the equivalent system x (m) 1 = f(t, x1, x 0 1 ,··· , x (m−1) 1 , x2, x 0 2 ,··· , x (n−1) 2 ), x (n) 2 = g(t, x1, x 0 1 ,··· , x (m−1) 1 , x2, x 0 2 ,··· , x (n−1) 2 ). Set u1 = x1, u2 = x 0 1 , u3 = x 00 1 ,··· , um = x (m−1) 1 , um+1 = x2, um+2 = x 0 2 , um+3 = x 00 2 ,··· , um+n = x (n−1) 2 . Linear Systems of Differential Equations