Find the integrating factor of the following first order ODE: dx/dt = -2tx +t.

Firstly rearrange the differential equation to fit the form dx/dt +P(t)x = Q(t). The integrating factor is then found by using the formula:

u = EXP(INTG(P(t))). We know that P(t) = 2t and so by integrating we find that INTG(P(t)) (which means the integral of P(t)) is equal to t². And so our integrating factor u = e^{t^2}.

Note: this is used in solving first order differential equations; by multiplying each term by the integrating factor and then some clever observation, you will see that the equation will now resemble the product rule formula. This can be used to solve the first order ODE as you will see in future questions.