Cosider the system of differential equationsx_1'=2x_1+x_2x_2'=4x_1-x_2Convert this system to a second order differential equations and solve this second order differential equations

Amari Flowers 2021-01-27 Answered

Cosider the system of differential equations
x1=2x1+x2
x2=4x1x2
Convert this system to a second order differential equations and solve this second order differential equations

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Expert Answer

sovienesY
Answered 2021-01-28 Author has 89 answers

Consider the given system f equations as {x1=2x1+x2x2=4x1x2
Solve the first equation x1=2x1+x2 for x2 and obtained as x2=x12x1
Substitude x2=x12x1 in the equation x2=4x1x2
x2=4x1x2
(x12x1)=4x1(x12x1)
x12x1=4x1x1+2x1
x1x16x1=0
Thus the second order differential equation is x1x16x1=0
Consider the equation x1x16x1=0
The characteristics equation of the second order differential equation is
m2m6=0
Solve the equation m2m6=0 as follows.
m2m6=0
m23m+2m6=0
m(m3)+2(m3)=0
(m3)(m+2)=0
m3=0 or m+2=0
m=3 or m=2
Thus, the general solution of the second order differential equations is obtained as below.
y=c1em1x+c2em2x
=c1e2x+c2e3x

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