Assume that y Ay has solutions y1 et 2e6t et y2 0 et 2et

Assume that y\' = Ay has solutions y_1 = (e^t 2e6t e^t), y_2 = (0 e^-t 2e^-t), y_3 = (e^2t e^2t 0). Show that the y_i\'s are linearly independent. Find the solution that satisfies y(0) = (0, 2, 3)^T.

Solution

Let, ay1+by2+cy3=0

So, ae^t+ce^{2t}=0

Hence, a=c=0

Hence, b=0 as y2 is non zero

So yi \'s are linearly independent

General solution is

y=Ay1+By2+Cy3

y(0)=(0,2,3)^T=A(1 ,2,1)^T+(0,1,2)^T+C(1,1,0)^T

A+C=0

2A+B+C=2

A+2C=3

HEnce, C=3, A=-C=-3

2A+B+C=2

B=5