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Hello! My name is Mikyla Smith and I currently attend Mississippi State University while majoring in Mechanical Engineering. I am excited to share my knowledge in math with you!

Show that $ y = \frac{2}{3}e^x + e^{-2x} $ is a solution of the differential equation $ y^x + 2y = 2e^x $.

Verify that $ y = -t \cos t - t $ is a solution of the initial value problem.

$ t \frac{ty}{dt} = y + t^{2} \sin t $ $ y = \pi = 0 $

(a) For what values of $ r $ does the function $ y = e^{rx} $ satisfy the differential equation $ 2y^{"} + y^{'} - y = 0? $

(b) if $ r_1 $ and $ r_2 $ are the values of $ r $ that you found in part (a), show that every member of the family of functions $ y = ae^{r_1{x}} + be^{r_2{x}} $ is also a solution.

(a) For what values of $ k $ does the function $ y = \cos kt $ satisfy the differential equation $ 4y^{"} = - 25y? $

(b) For those values of $ k. $ verify that every member of the family functions $ y = A \sin kt + B \cos kt $ is also a solution.

Which of the following functions are solutions of the differential equation $ y^{"} + y = \sin x ? $

(a) $ y = \sin x $(b) $ y = \cos x $(c) $ y = \frac {1}{2} x \sin x $(d) $ y = - \frac{1}{2} x \cos x $

DISCUSS: An Algebra Error Beginning algebra studentssometimes make the following error when squaring a binomial:$$(x+5)^{2}=x^{2}+25$$(a) Substitute a value for $x$ to verify that this is an error.(b) What is the correct expansion for $(x+5)^{2} ?$