Please help We do not need to use matrices on this problem T

Please help. We do not need to use matrices on this problem.

This is from differena equations: Solve the i l-value problem dt = 16x-y dt x(1) = 0, y(1) = 1

x\'=-9x-y

y\'=16x-y

From first equation

y=-9x-x\'

y\'=-9x\'-x\'\'=16x-y=16x+9x+x\'

x\'\'+10x\'+25x=0

Let, x=exp(kt)

Substituting gives

k^2+10k+25=0

k=-5

x(t)=exp(-5t)(A+Bt)

y=-9x-x\'

x\'=exp(-5t)(-5A+B-5Bt)

y=-9exp(-5t)(A+Bt)-exp(-5t)(-5A+B-5Bt)

y(t)=exp(-5t)(-4A-B+4Bt)

x(1)=exp(-5)(A+B)=0

So, A=-B

y(1)=exp(-5)(-4A-B+4B)=1=exp(-5)(-4A+A-4A)=1

A=-exp(5)/7

B=exp(5)/7