What a differential equation is doing

A differential equation links an unknown function with one or more of its derivatives. The usual form in this topic is

A first-order differential equation has the first derivative as its highest derivative. For example, $\frac{dy}{dx}=x+y^2$ is first order. An equation containing $\frac{d^2y}{dx^2}$ does not belong to this syllabus point.

A solution is a function that satisfies the differential equation on an interval. A general solution describes a family of solutions and contains an arbitrary constant. Extra information selects one member of that family, called a particular solution. An initial condition gives the value of the unknown function at one input. For example,

The three analytic types in this topic

IB expects you to recognize and solve three common first-order types:

• separable equations, where the $x$-parts and $y$-parts can be separated before integrating;
• homogeneous equations, where the right-hand side depends on the ratio $\frac{y}{x}$;
• first-order linear equations, which an integrating factor can turn into the derivative of a product.

The topic also includes one numerical method: Euler's method. It approximates a solution when an exact solution is not required or is inconvenient to find.

In modelling, the derivative tells the rate story. Newton cooling models temperature change, while population models describe growth rate. Carbon dating uses radioactive decay; physics uses decay curves, and chemistry uses first-order reaction curves. The same mathematical structure appears in each case, but the variables have different meanings. There is a TOK point here too. Personal experience may suggest which kind of model feels plausible, but the mathematical knowledge claim depends on whether the equation and its solution follow logically from the assumptions.

The tangent-line idea

Euler's method is a numerical method for approximating the solution curve of a first-order differential equation. Starting at a known point, it follows a short tangent-line segment, then repeats the process. The derivative at the current point gives the gradient for the next small step.

For

Euler's method uses

and

A step length is the fixed increment in the independent variable that takes the method from one approximation point to the next. A smaller $h$ usually gives a better approximation, though it requires more repeated calculations. In an exam, use the given $h$. Don't choose your own unless the question asks you to.

How to lay out the method

Begin with the given initial condition. This is usually written as

and $y_0=b$

Then substitute the current pair into the recurrence for the problem.

For example, if

then the iteration is

and

Show the first step by hand:

After that, it makes sense to use a calculator table or spreadsheet. Check that the formula entered matches the differential equation exactly. Pay particular attention to the brackets around $f(x_n,y_n)$.

Interpreting the approximation

Euler's method doesn't produce the exact solution. Instead, it creates a polygonal path with gradients that copy the differential equation at the left-hand end of each step. The approximation can drift away quite quickly when the true solution bends sharply or $h$ is large. The method is useful, but it isn't magic.

Recognising a separable equation

A separable differential equation is a first-order differential equation that can be rearranged to place every term involving $y$ with $dy$ and every term involving $x$ with $dx$. It often has the form

As long as the division is valid on the interval under consideration, rearrange the equation as

Then integrate both sides:

Here, $C$ is the arbitrary constant of integration. Don’t lose it. When an initial condition is given, substitute it after integrating.

A clean separable example

Consider

First separate the variables, then integrate:

which gives

Now use $y(1)=1$:

and hence

Therefore,

so

Take care with the final algebra. From $y^2=x^2+C$ you get $y=\pm\sqrt{x^2+C}$ unless the context or initial condition determines the sign. Don’t casually turn a square into a linear expression.

Logistic growth and partial fractions

A syllabus example is the logistic equation. This separable differential equation models a quantity that grows quickly at first, then slows as it approaches a limiting value:

The value $a$ is the carrying capacity, the limiting quantity approached by the model as growth slows. Separating the equation gives

Partial fractions now become useful. Partial fraction decomposition is an algebraic method for rewriting a rational expression as a sum of simpler rational expressions. For $a\ne 0$,

Thus,

Integrating gives

An initial condition can then be used to find $C$. If the question asks for $n$ explicitly, rearrange the result for $n$.

Recognising the type

A homogeneous first-order differential equation is a first-order differential equation in which the right-hand side depends on $x$ and $y$ only through the ratio $\frac{y}{x}$. In the syllabus, it has the form

Here, homogeneous refers to scaling. To check, replace $x$ with $tx$ and $y$ with $ty$, where $t$ is a non-zero scale factor. The equation is homogeneous if the $t$ cancels completely from the right-hand side.

For example,

is homogeneous since

so the right-hand side depends only on ratios.

The substitution that unlocks it

A substitution deliberately replaces variables to put an equation into a more useful form. For homogeneous equations, use

Differentiate $y=vx$ with respect to $x$:

The substitution should turn the equation into a separable one. That’s the purpose of the method.

Typical working pattern

Start with the example above:

Put $y=vx$. Then $\frac{y}{x}=v$ and $\frac{x}{y}=\frac{1}{v}$, giving

Subtract $v$ from both sides:

Now separate the variables:

Integrate:

Finally, replace $v$ with $\frac{y}{x}$:

If the question states $x>0$, then $\ln|x|$ can be written as $\ln x$. It’s easy to forget the final substitution, so don’t leave the answer in terms of $v$ unless the question explicitly allows it.

The form to look for

A first-order linear differential equation is a first-order differential equation where the unknown function and its first derivative appear only to the first power. They aren’t multiplied together. Its standard form is

Before choosing the integrating factor, put the equation into this exact form. If $y'$ has a coefficient in front of it, divide through first.

Integrating factor

An integrating factor is a non-zero function that multiplies a differential equation, turning the left-hand side into the derivative of a product. For

use

We normally omit the constant of integration inside the exponential since any non-zero constant multiple of an integrating factor works.

Now multiply the entire equation by $\mu(x)$:

The left-hand side has been constructed so that

Integrate both sides:

If the question asks for $y$ as a function of $x$, finish by dividing by $\mu(x)$.

Example with the product derivative visible

Solve

In this equation,

So the integrating factor is

On an interval where $x>0$, this becomes

Multiply every term by $x^{-5}$:

The expression on the left is

so

Now integrate:

Therefore,

The product-derivative check isn’t just decoration. It’s the reason the method works.

Where this model appears

First-order linear equations often appear when a rate is proportional to a difference or to the current amount. Newton's law of cooling, for example, can model a temperature approaching room temperature. Carbon dating and physics decay curves use exponential decay ideas. In chemistry, first-order reactions have rates proportional to the amount of reactant remaining. Although the contexts differ, the same mathematical structure describes change. It also lets us compare patterns and make predictions from rates.