Could you please solve the differential equations Transform

Could you please solve the differential equations?

Transform the (linear) differential equation x^(3) = t^3 - t^2 x + cos (2t) x\' - x\" into an equivalent system of first-order (linear) differential equations.

Solution

We need to convert the given third order differential equation into a system of three first order differential equations

Let,

x\'=v

x\'\'=(x\')\'=v\'=w

Hence

x\'=v

v\'=w

x\'\'\'=t^3-t^2x+cos(2t)x\'-x\'\'

(x\'\')\'=t^3-t^2x+cos(2t)v-w

w\'=t^3-t^2x+cos(2t)v-w