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**Quiz-4 / Quiz 4 (TUT0601)**

« **on:**October 18, 2019, 10:47:11 PM »

Find the general solution of this equation:

Find the homogenous solution:

Then,

Now, consider

Want to find:

Then,

Thus, solving the equation, we get that:

Now, consider

Want to find:

Repeat for the process for Y(t) and solve for B. Hence, we get that

Lastly,

y''+y'-6y = 12e

^{3t}+12e^{-2t}Find the homogenous solution:

y''+y'-6y=0

r

^{2}+r-6r=0(r+3)(r-2)=0

r

_{1}= -3, r_{2}= 2y

_{1}(t)= e^{-3t}, y_{2}(t)= e^{2t}Then,

y(t) =c

_{1}e^{-3t}+ c_{2}e^{2t}Now, consider

y''+y'-6y = 12e

^{3t}Want to find:

Y(t) = Ae

^{3t}Y'(t) = 3Ae

^{3t}, Y''(t) = 9Ae^{3t}Then,

9Ae

^{3t}+ 3Ae^{3t}- 6Ae^{3t}= 12e^{3t}Thus, solving the equation, we get that:

A = 2 and Y(t) = 2e

^{3t}Now, consider

y''+y'-6y = 12e

^{-2t}Want to find:

Y(t) = Be

^{-2t}Repeat for the process for Y(t) and solve for B. Hence, we get that

B = -3 and Y(t) = -3e

^{-2t}Lastly,

y = c

_{1}e^{-3t}+ c_{2}e^{2t}+ 2e^{3t}- 3e^{-2t}