Differential equation?

Question

The solution of a certain differential equation is of the form

y(t) = ae^(7 t) + be^(12 t),

where a and b are constants.

The solution has initial conditions y(0) = 2 and y'(0) = 5 .

Find the solution by using the initial conditions to get linear equations for a and b.

can you please help?

Answer 1

to use the initial conditions starting with y at time 0 =2, we get

y(0)=ae^0+be^0

1) a+b=2

Next we take the derivative of y with respect to t dy/dt = y'

2) y' = 5=7*ae^(7t)+12*be^(12t) @t=0

        5=7a+12b

Now we solve 1) for a=2-b and substitute into 2)

7(2-b)+12b=5

b=-9/5

a=b-2 =-9/5-10/5= -19/5

Now we integrate the differential equation

Integral[y(t)]=a/7 * e^7t +b/12 * e^12t = -19/35*(e^7t) -9/60 * (e^12t).

The solution can be further simplified by factoring out certain terms, such that

Integral [y(t)] = -1/5 * e^7t * [19/7 +9/12 * e^5t]