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37)

a) Write the initial value problem in vector form. b) Solve by using eigenvalues/eigenvectors. c) Write the solution in $x(t),y(t)$ form. 36. $dtdx =2x+ydtdy =3x+4y x(0)=3y(0)=1 $ 37. $dtdx =3x−2yx(0)=2dtdy =4x−y,y(0)=−1 $

a) To write the initial value problems in vector form, we introduce a vector function:

For problem 36, the system of differential equations can be written in vector form as:

with the initial condition:

Similarly, for problem 37, the system of differential equations can be written in vector form as:

with the initial condition:

b) To solve these systems using eigenvalues and

eigenvectors, we first find the eigenvalues and

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