Rewrite the following differential equation as an equivalent

Rewrite the following differential equation as an equivalent first-order system of differential equations:

d²y/dt² + 4e^tdy/dt 9t²y = 7t².

Solution

Differential equation is

y\'\' + 4e^(t)y\' - 9t^2y = 7t^2

x1(t) = y(t), x2(t) = y\'(t)

x1\'(t) = x2(t)

x2\'(t) + 4e^(t)x2(t) - 9t^2*x1(t) = 7t^2

Hence the two differential equation of first order will be

x1\'(t) = x2(t)

x2\'(t) + 4e^(t)x2(t) - 9t^2*x1(t) = 7t^2