Question

1. (3 pts) Solve the initial value problem
   25y′′−20y′+4y=0, y(5)=0, y′(5)=−e2.

2. (3 pts) Solve the initial value problem
   y′′ − 2√2y′ + 2y = 0, y(√2) = e2, y′(√2) = 2√2e2.

3. Consider the second order linear equation t2y′′+2ty′−2y=0, t>0.

   1. (a) (1 pt) Show that y1(t) = t−2 is a solution.

   2. (b) (3 pt) Use the variation of parameters method to obtain a second solution and a general solution.

Answer 1

summary:- For que 1 we first simolify given differential equation in standard form. Then we write its auxillary equation to find roots of eq. We see roots are equal then we write its solution and by initial value problem we find value of arbitary constant. Then put value of constant and we get solution of differential eq. Same process follow in que. 2.

For que 3 part (a) we check given solition satisfied or not given diff. eq. For part (b) we use cauchy euler equation to solve part(b).