Mastering Mathematics Smartly
by Wee Wen Shih

A unique self-help website that provides comprehensive coverage of mathematics at A-level & beyond, written in a student-friendly style.

Integrating factor method

The integrating factor method is not covered in H2 mathematics but it is examined in H3 mathematics.

The method is useful in solving first-order differential equations of the form \frac{dy}{dx} + Py = Q where P, Q are functions of x alone.

Suppose we wish to solve cos x\frac{dy}{dx} + y = sin x. We follow the following steps systematically.

Step 1: Write the DE in the standard form.

We obtain \frac{dy}{dx} + \frac{1}{cos x}y = tan x.

Step 2: Find the integrating factor of the form e^{\int_{}^{} P\: dx}.

We obtain
e^{\int_{}^{} sec x\: dx} \\= e^{ln |sec x + tan x|} \\= sec x + tan x \\= \frac{1 + sin x}{cos x}


Step 3: Multiply both sides of the DE by the integrating factor.

We obtain \left(\frac{1 + sin x}{cos x}\right)\left(\frac{dy}{dx} + \frac{1}{cos x}y\right) = \left(\frac{1 + sin x}{cos x}\right)tan x,

which can be expressed as \frac{d}{dx}\left[\left(\frac{1 + sin x}{cos x}\right)y\right] = \left(\frac{1 + sin x}{cos x}\right)tan x.

Step 4: Integrate both sides with respect to x.

We obtain \left(\frac{1 + sin x}{cos x}\right)y = \int_{}^{}\left(\frac{1 + sin x}{cos x}\right)tan x\:dx
\\= \int_{}^{}(sec x\:tan x + tan^2 x)\:dx \\= \int_{}^{}sec x\:tan x\:dx + \int_{}^{}(sec^2 x - 1)\:dx \\= sec x + tan x - x + c

Finally, we have y = 1 - \frac{x\:cos x}{1 + sin x} + \frac{c\:cos x}{1 + sin x} as a general solution of the DE when we express y in terms of x.

Practice: Apply the integrating factor method to solve tan x\:\frac{dy}{dx} + y = sec x.