How to use the substitution u = y(x) in a separable differential equation

Key concept: This step often looks more mysterious than it is. The point is simply to rewrite the left side in a way that matches the derivative already present in the equation.

Step by step

When you have a separable equation in the form $N(y)\frac{dy}{dx}=M(x),$ you can multiply by $dx$ and write $N(y)\,dy=M(x)\,dx.$

At that point, the substitution $u=y(x),\qquad du=y'(x)\,dx=\frac{dy}{dx}\,dx$ just restates the same idea in standard calculus notation.

So the integral becomes $\int N(u)\,du=\int M(x)\,dx.$

That is the key move:

• left side: integrate with respect to $u$,
• right side: integrate with respect to $x$.

If the original equation came from separating variables, this is exactly what you want. After integrating, replace $u$ with $y$ again and solve for the solution if possible.

Pitfall alert

Don’t write $du = dy$ unless you have clearly set up the dependence on $x$. The safer identity here is $du = y'(x)\,dx = \frac{dy}{dx}\,dx$. Also, make sure the differential on each side matches the variable of integration; mixing $dx$ and $dy$ in the same integral is a red flag.

Try different conditions

If the left-hand side is something like $\int y\,dy$, the substitution is almost automatic and you may not need to rename the variable at all. If the equation includes an initial condition, plug it in after integrating to determine the constant before solving for $y(x)$.

Further reading

variable separation, integration substitution, implicit solution