Find a particular solution to the given system and evaluate

Find a particular solution to the given system and evaluate the function.

x\'=y

y\'=16x

x(0)=10

y(0)=10

Find x(ln(2))

Solution

x\'=y

y\' = x\'\' = 16x

equation

x\'\' -16x = 0

x(0) = 10

x\'(0) = 10

general solution of this equation is

c1 e^4x + c2 e^(-4x)

x(0) = 10 gives c1 + c2 = 10

x\'(0) = 10 gives 4c1 - 4c2 = 10

so 8c1 = 5 0 = > c1 = 50/8

c2 = 30/8

x(ln2) = c1 * 2^4 + c2 * 2^-4 = 100 + 30/128