Question

(1 point) Math 216 Homework webHW9, Problem 4 Use Laplace transforms to find a nontrivial solution to Require that x1) to find any arbitrary constant in your solution.

Answer 1

" Differential equation t x"" t (2t - 6) x' + 2x = 0 x(0) = 0 & x(1) = e^-2 taking Laplace transform on both side of the given equation we have: {t x""} + 2 L {t x'} - 6 L {x'} + 2 L [x} = 0 -d/ds {s^2 x - s x(0) - x' (0)} + 2[-d/ds {s x - x (0)} - 6 {s x - x(0)} + 2 x = 0 Rightarrow -d/ds {s62 x - x' (0)} - 2d/ds {s x} - 6 {s x} + 2 x = 0 because x (0) = 0 - s^2 dx/ds - 2s x + 0 - 2 s dx/ds - 2 x - 6 s x + 2x = 0 - s^2 dx/ds - 2 s dx/ds = theta s x dx/ds = theta s x/-s^2 - 2s dx/x = -theta/s + 2 ds on integrating we have log x: - theta log (s + 2) x = 1/(s = 2)^theta "