How to solve a separable differential equation by substitution

Expert intro: A separable differential equation is one of those problems where the setup matters more than the algebra. Once the variables are split correctly, the substitution step is usually simple.

Detailed walkthrough

If the differential equation has the separable form $\frac{dy}{dx}=\frac{M(x)}{N(y)},$ then we can rewrite it as $N(y)\,dy=M(x)\,dx.$

Now integrate both sides: $\int N(y)\,dy=\int M(x)\,dx.$

If your notes write the left side using $y=y(x)$, then the substitution $u=y(x),\qquad du=y'(x)\,dx=\frac{dy}{dx}\,dx$ just formalizes the idea that $y$ depends on $x$.

So the workflow is:

1. Separate the variables so all $y$ terms are on one side and all $x$ terms are on the other.
2. Integrate both sides.
3. Use the substitution $u=y(x)$ if needed to make the left integral easier to read.
4. Solve for $y$ if an explicit solution is possible.

For example, if $N(y)\,\frac{dy}{dx}=M(x),$ then multiplying by $dx$ gives $N(y)\,dy=M(x)\,dx,$ and after integration you get $\int N(y)\,dy=\int M(x)\,dx+ C.$ That is the standard separable-equation pattern.

💡 Pitfall guide

A frequent mistake is treating $y$ like a constant when integrating the left-hand side. If $y=y(x)$, then you cannot ignore the dependence on $x$; that is exactly why the substitution $u=y(x)$ is helpful. Another common issue is forgetting the constant of integration after integrating both sides.

🔄 Real-world variant

If the equation is not separable, this method does not apply directly. In that case you may need an integrating factor, a substitution, or a numerical method. If an initial condition such as $y(x_0)=y_0$ is given, use it right after integrating to determine the constant $C$.

🔍 Related terms

separable differential equation, substitution, constant of integration