# Understanding the integrating factor

I'm trying to understand solving ordinary linear differential equations by using integrating factor. I dont have my book yet so I'm reading from wikipedia https://en.wikipedia.org/wiki/Integrating_factor#Use_in_solving_first_order_linear_ordinary_differential_equations but I really don't get it.

I have my equation $y' + P(x)y = Q(x)$ , I multiply on both sides with my integrating factor $e^{\int{P(x)}dx}$. If I now integrate on both sides with respect to x and then I'm magically seems to get something simpler.

Can someone please help me understand whats happening in the transformation of the left hand side of the equation when it goes from a partial to a total derivative? Why does this work?

It works because exponential function has special property. By chain rule for function $f(x)$ you have:

$$(e^{f(x)})'=f'(x)e^{f(x)}$$

It's a very commonly used trick, so you should remember it. If you also use formula $(fg)'=fg'+f'g$ you get your "magic". For example:

$$y'(x)+2y(x)=0$$

Multiply both sides by $e^{2x}$ to get something like $(fg)'=fg'+f'g$:

$$y'(x)e^{2x}+2y(x)e^{2x}=y'(x)\cdot e^{2x}+y(x)\cdot (e^{2x})'=(y(x)e^{2x})'=0$$.

• Thank you very much I really appreciate it. I get it now =) Commented Sep 2, 2014 at 4:41

The answer by agha already explains why the "magic" works. In this answer, I will further elaborate why the method of using integrating factor simplifies the LHS of a differential equation of the form $$y' + yP(x) = Q(x)$$ so well.

Consider the linear first-order differential equation $$y' + yP(x) = Q(x). \tag{1}\label{eq1}$$ We first find the integrating factor $$M(x) = e^{\int P(x)\, dx}. \tag{2}\label{eq2}$$

### An Interesting Relationship

Now this integrating factor satisfies an interesting relationship $$M'(x) = M(x) P(x). \tag{3}\label{eq3}$$ We can prove this relationship easily by differentiating both sides of \eqref{eq2} as follows: $$M'(x) = \frac{d}{dx} \left( e^{\int P(x)\, dx} \right) = e^{\int P(x)\, dx} \frac{d}{dx} \left( \int P(x)\, dx \right) = M(x) P(x).$$ Note that we use the chain rule to work out the derivative above. This interesting relationship will come useful.

### Reducing the LHS to a Single Derivative

Now let us multiply the integrating factor $$M(x)$$ on both sides of the differential equation \eqref{eq1}. By doing so, we get $$y' M(x) + y P(x) M(x) = Q(x) M(x).$$ But from \eqref{eq3} we know that $$P(x) M(x) = M'(x)$$, so the above equation can be written as $$y' M(x) + y M'(x) = Q(x) M(x).$$ Look what has happened on the left hand side. On the left hand side we have the expansion of $$\frac{d}{dx}(yM(x))$$, that is, by product rule we have $$\frac{d}{dx}(yM(x)) = y' M(x) + y M'(x)$$. Therefore the above equation can be written as $$\frac{d}{dx}(yM(x)) = Q(x) M(x).$$ The "magic" has occurred here! Multiplying both sides of the differential equation has led us to an equation that has got a single derivative only on the LHS. So now finding the solution $$y(x)$$ is a simple matter of integrating both sides of the previous equation, i.e., $$y M(x) = \int Q(x) M(x) \, dx.$$ Thus $$y = \frac{1}{M(x)} \int Q(x) M(x) \, dx.$$ Note that the result of indefinite integral on the RHS will contain the constant of integration, which we will denote as $$C$$, so the result would look like $$y = \frac{1}{M(x)} \int Q(x) M(x) \, dx + \frac{C}{M(x)}.$$

### Illustration

Let us illustrate the method with a very simple differential equation: $$y' + \frac{y}{x} = x.$$ First we note that this is indeed in the form $$y' + yP(x) = Q(x)$$ with $$P(x) = 1/x$$ and $$Q(x) = x$$. We now compute the integrating factor $$M(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\ln x} = x.$$ Then we multiply the integrating factor on both sides of the differential equation to get $$y'x + y = x^2.$$ Now indeed the LHS can be written down as a single derivative as shown below: $$\frac{d}{dx} yx = x^2.$$ Note that the LHS is the derivative of the product of $$y$$ and the integrating factor (exactly as shown in the previous section). Now we integrate both sides, $$yx = \frac{x^3}{3} + C.$$ Finally we divide both sides by the integrating factor (again, exactly as shown in the previous section). The integrating factor for this problem happens to be $$x$$, so we divide both sides by $$x$$ to get $$y = \frac{x^2}{3} + \frac{C}{x}.$$ We have arrived at the solution $$y(x)$$ for the differential equation.