thanksSolutionThe equation can be rewritten as dydt 18y 8

Solution

The equation can be re-written as :

dy/dt + (1/8)y = 8t

which is a linear differential equation.

Integrating factor (IF)= e^{integral 1/8 dt}

= e^t/8

The solution of the equation is of the form :

y * IF = (integral RHS * IF dt) + c

y * e^t/8 = integral ( 8t * e^t/8 ) + c

y * e^t/8 = 8 * integral ( t * e^t/8 ) + c

Using integration by parts to solve the integral, we get,

y * e^t/8 = 8 * [8e^t/8 (t - 8)] + c

y * e^t/8 = 64e^t/8 (t - 8) + c

y(t) = 64(t - 8) + ce^-t/8

Now y(0) = 3

3 = 64(0-8) + c

c = 515

y(t) = 64(t - 8) + 515e^-t/8