# Integrating both sides of an equation with respect to what?

Let's say we have the following DE:

$$\frac{dy}{dx} = x$$

1. That's how It could be solved:

$$dy = x dx$$

$$\int{dy}=\int{xdx}$$

$$y=\frac{1}{2}x^2+c$$

Is it mathematically correct to separate $$dy$$ and $$dx$$ ? Or it would appear as the derivative of $$y$$ with respect to $$x$$ is a fraction which is not true.

2. Another method of writing the solution can be:

$$\int\frac{dy}{dx} dx=\int{xdx}$$

$$\int{dy}=\int{xdx}$$

$$y=\frac{1}{2}x^2+c$$

That's integrating the DE with respect to x. Is this mathematically correct? It seems like the $$dx$$ cancels each other in the LHS, but we are not allowed to treat this as a fraction.

My Questions are:

• Which of these ways of writing the solution is the most accurate?
• And which of these 2 is wrong in terms of following mathematical structure?
• When we integrate both sides of the equation, can we just put the integral sign without integrating with respect to a variable? (for example in case 1)
• Third method should be $\int y'\,dx.$ If you made that change, all three methods are equally valid. – Adrian Keister Oct 1 '19 at 17:12
• It is probably more correct to write $\int {dy(x) \over dx} dx = \int x dx$... – copper.hat Oct 1 '19 at 17:16
• To justify the first method in full rigor, one needs to develop the theory of differential forms. But the upshot is that, all the manipulations in your solution can be validated. – Sangchul Lee Oct 1 '19 at 17:25
• The integral sign is very important as it defines the integration. It also tells us if it indefinite or definite. – Sam Oct 1 '19 at 18:00
• – Hans Lundmark Oct 2 '19 at 4:03

The third is wrong. $$\int y'dy \ne y$$, since $$y'\ne 1$$.
It is separable equation where $$y= y(x)$$. Consider the problem $$\frac{dy}{dx}=F(x)*Q(y)$$ where $$F(x)$$ depends only of $$x$$ and $$Q(y)$$ only of $$y$$. If $$Q(y) \neq 0$$ we can rewrite this as $$\frac{y'(x)}{Q(y(x))}= F(x)$$ $$\int_{x_0}^x{\frac{y'(t)}{Q(y(t))}dt}=\int_{x_0}^x{F(t)dt}$$ Then $$y(t)=s$$ We will get $$\int_{y(x_0)}^{y(x)}{\frac{ds}{Q(s)}} = \int_{x_0}^x{F(t)dx}$$
When your diffferential equation is separable you end up with a differential form $$f(y)dy = g(x) dx$$
Since you are assuming taht $$y=y(x)$$ you may integrate both sides with respect to $$x$$ but the LHS has gone through a change of variable so we get $$\int f(y(x))y'(x)dx = \int f(y)dy = \int g(x) dx$$