Substituting the given vector into the system and simplifying the resulting equations, we find that both equations hold true for any value of t. Therefore, the vector is indeed a particular solution of the given nonhomogeneous linear system.
My pleasure, I’ve been growing my expertise in solving system of linear differential equations problems. Let's verify that the vector xp, is a particular solution of the given nonhomogeneous linear system.
dx / dt =x+4y+2t−5;
dy / dt =3x+2y−4t−12;xp =( 2 −1 )
We can verify that xp is a solution by substituting it into the system and checking if the equation is satisfied.
Steps to solve:
1. Substitute xp into the system:
d / dt (2)=−1+4(−1)+2t−5
d / dt (−1)=3(2)+2(−1)−4t−12
2. Simplify the equations:
d / dt(2)=−7+2t−5
d /dt (−1)=6−4t−12
3. Check if the equations are satisfied:
The first equation becomes −7+2t−5= −7 +2t−5, which is always true.
The second equation becomes 6−4t−12= 6 −4t−12, which is also always true. Therefore, we can conclude that xp is a particular solution of the given nonhomogeneous linear system.