How to Solve Differential Equations

A separable differential equation is one you can split into an $x$ part and a $y$ part, integrate each side, and solve. It is usually the first type students learn, and the calculation is short, but the care lives in where you divide and what conditions that division requires. Here is the formula, why it works, and a full worked solution.

The Setup And Its Symbols

A differential equation contains an unknown function $y$ and its derivative $\frac{dy}{dx}$, and the solution is a function, not a single number. For example, equations like

are differential equations. What we want is a function $y$ that satisfies the equation. For instance, $y = x^2 + C$ satisfies $\frac{dy}{dx} = 2x$, and an initial condition fixes $C$ to give one specific solution.

A separable equation has the form

so that it can be rewritten as

and integrated side by side.

Why Separation Works

The method rests on a single idea: if you can get everything involving $y$ onto one side and everything involving $x$ onto the other, then integrating both sides with respect to their own variables undoes the derivative. Writing $\frac{1}{h(y)}\,dy = g(x)\,dx$ is shorthand for that move, and $\int \frac{1}{h(y)}\,dy = \int g(x)\,dx$ recovers a relationship between $y$ and $x$ up to a constant. The catch is the division: rewriting requires dividing by $h(y)$, which is only valid where $h(y) \ne 0$. Each $y$ with $h(y)=0$ gives a constant solution that the division step quietly drops, so those have to be checked separately.

The Procedure For Separable Equations

1. Confirm the equation is in the form $\frac{dy}{dx} = g(x)h(y)$.
2. Separate the $y$ part and the $x$ part.
3. Integrate both sides.
4. Simplify exponents or logarithms to express $y$.
5. Use the initial condition to determine the constant of integration.

More important than calculation speed is staying aware of where you divided and under what conditions that transformation is valid.

Worked Example: $\frac{dy}{dx} = 2xy,\ y(0)=3$

The right side is $2x \cdot y$, so this is separable. Given $y(0)=3$, we can assume $y \ne 0$ at least near $x=0$. In that range, dividing by $y$ gives

Integrate both sides:

Convert to exponential form:

Absorbing the sign of the constant, $y = Ce^{x^2}$. Applying $y(0)=3$:

The takeaway is not the calculation but whether the variables could be separated and what the condition for dividing by $y$ was. Note that $y=0$ is also a solution of $\frac{dy}{dx}=2xy$, but it does not satisfy $y(0)=3$.

Direct Practice With A Check

Solve the following with the same procedure:

Separate $y$ to the left and $x$ to the right, integrate, and use the initial condition. Self-check: you should reach $\ln|y| = \tfrac{3}{2}x^2 + C$, then $y = 2e^{\frac{3}{2}x^2}$, and substituting $x=0$ must return $y=2$. Each time you try another problem, verify where you divided and whether any solution was lost.

Calculation Traps

1. Applying these steps to an equation that is not separable.
2. Forgetting the constant of integration $C$ after integrating both sides.
3. Being sloppy with absolute values or constant absorption when converting $\ln|y|$ to $y$.
4. Dividing by $y$ or $h(y)$ without checking whether those values could be $0$.
5. Stopping at the general solution without using the initial condition to find the specific one.

The fourth is the most dangerous: even with clean calculations you might miss a valid solution. Whenever you divide, double-check whether a solution exists when that term is $0$.

Where Differential Equations Are Used

They appear whenever we know the law governing a change and want the original function back: finding position from velocity, building growth and decay models, or tracking electric current or chemical concentration. Not every differential equation is separable, though. First-order linear, homogeneous, and second-order equations each call for different tools, which is why identifying the type comes before the calculation.

One Last Verification Habit

Mastering separable equations means mastering the verification steps as much as the integration. Before you call a problem done, confirm the constant is determined, the absolute value was handled cleanly, and no constant solution was dropped along the way.