Please help with this Differential equations problem Show th

Please help with this Differential equations problem

Show that any function of the form

satisfies the differential equation

Solution

x = Acosh(wt) + Bsinh(wt)

Deriving :
x\' = Ad/dt(cosh(wt)) + Bd/dt(sinh(wt))
x\' = Asinh(wt)*w + B*cosh(wt)*w
x\' = Awsinh(wt) + Bwcosh(wt)

Deriving again to find x\'\' :
x\'\' = Aw*d/dt(sinh(wt)) + Bw*d/dt(cosh(wt))
x\'\' = Aw*cosh(wt)*w + Bw*sinh(wt)*w
x\'\' = Aw^2cosh(wt) + Bw^2sinh(wt) ---> First blank

Now, w^2x becomes :

w^2 * [Acosh(wt) + Bsinh(wt)]
Aw^2*cosh(wt) + Bw^2*sinh(wt) --> Second blank

Now, finally x\'\' - w^2x :

Aw^2cosh(wt) + Bw^2sinh(wt) - (Aw^2*cosh(wt) + Bw^2*sinh(wt))

Aw^2cosh(wt) + Bw^2sinh(wt) - Aw^2cosh(wt) - Bw^2sinh(wt)

0 ---> Third blank