Differential Equations : Nonhomogeneous Linear Systems

Study concepts, example questions & explanations for Differential Equations

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Nonhomogeneous Linear Systems

Solve the following system.  

Possible Answers:


Correct answer:


First, we will need the complementary solution, and a fundamental matrix for the homogeneous system. Thus, we find the characteristic equation of the matrix given.

Using , we then find the eigenvectors by solving for the eigenspace.

This has solutions , or . So a suitable eigenvector is simply .

Repeating for ,

This has solutions , and thus a suitable eigenvector is . Thus, our complementary solution is  and our fundamental matrix (though in this case, not the matrix exponential) is . Variation of parameters tells us that the particular solution is given by , so first we find  using the inverse rule for 2x2 matrices. Thus, . Plugging in, we have . So  .

Finishing up, we have .

Adding the particular solution to the homogeneous, we get a final general solution of 

Learning Tools by Varsity Tutors