Solve the given system of differential equations. Q.x′_1 =x_1−3x_2, x′_2 =3x_1+x_2.

Bergen 2021-02-09 Answered
Solve the given system of differential equations.
Q.x1=x13x2,x2=3x1+x2.
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Expert Answer

timbalemX
Answered 2021-02-10 Author has 108 answers

Step 1
Provided a system of differential equations.
Arrange the system as shown below,
x'=Ax
x(t)=[x1(t)x2(t)]
A=[1331]
Eigen values of A is given by |AλI|=0
(1λ)2+9=0
(1λ)2=9
(1λ)=±3i
λ=±3i+1
Take λ=3i+1
AλI[13i13313i1]
AλI=[3i333i]
Step 2
Now
3ix13x2=0
3ix1=3x2
ix1=x2
Take Eigen vector v1=[1j]
Then x(t)=eλtv1=e3i+1v1
x(t)=e3i+1[1j]
x1(t)=[e3i+1ie3i+1]
As both the Eigen values are conjugate hence,
x2(t)=[e3i+1ie3i+1]

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