IB AA: Differential Equations

Differential equations are the language of change, translating dynamic processes in physics, biology, economics, and engineering into a form we can analyze and solve. In IB Mathematics Analysis & Approaches (AA), you move beyond simply finding derivatives to reversing the process: discovering the original functions that describe how quantities evolve. Mastering this topic equips you to model everything from population growth to the cooling of an object, providing a powerful toolkit for understanding a world in motion.

What is a Differential Equation?

A differential equation is an equation that relates a function with one or more of its derivatives. In this course, we focus on first-order ordinary differential equations (ODEs), which involve only the first derivative $y^{\prime }$ or $\frac{dy}{dx}$ of an unknown function $y(x)$. The order is determined by the highest derivative present. The core challenge is to find the general solution, a family of functions that satisfies the equation, often involving an arbitrary constant $C$. Applying an initial condition, like $y(0)=5$, allows you to find the particular solution that fits a specific scenario. Your goal is to move from a statement about the rate of change to a precise description of the quantity itself.

Core Method 1: Separable Variables

The first and often simplest technique is for separable variable differential equations. These are equations where you can algebraically manipulate all terms involving $y$ to one side and all terms involving $x$ to the other. The standard form is $\frac{dy}{dx}=g(x)h(y)$ or, more simply, $\frac{dy}{dx}=f(x)g(y)$.

The solution process is a three-step algorithm:

1. Separate: Rearrange the equation into the form $\frac{1}{g(y)}dy=f(x)dx$. Treat $\frac{dy}{dx}$ as a fraction for this purpose.
2. Integrate: Integrate both sides with respect to their own variable: $$\int \frac{1}{g(y)}dy=\int f(x)dx.$$
3. Solve and Simplify: Perform the integrations, include the constant of integration $C$ on one side, and then solve explicitly for $y$ if possible.

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

1. Separate: $\frac{1}{y}dy=2xdx$.
2. Integrate: $\int \frac{1}{y}dy=\int 2xdx\Rightarrow \ln |y|=x^2+C$.
3. Solve: $|y|=e^{x^2+C}=e^C{e}^{x^2}$. Let $A=\pm e^C$, so $y=A{e}^{x^2}$.
4. Apply Initial Condition: $3=A{e}^0\Rightarrow A=3$. The particular solution is $y=3e^{x^2}$.

Core Method 2: The Integrating Factor

Not all first-order ODEs are separable. A major class of non-separable equations are linear first-order equations, which can be written in the standard form $\frac{dy}{dx}+P(x)y=Q(x)$. The solution method here is the integrating factor method.

The strategy is to multiply the entire equation by a cleverly chosen function, called the integrating factor $I(x)$, which makes the left-hand side a perfect derivative. This factor is always defined as $I(x)=e^{\int P(x)dx}$.

Step-by-step procedure:

1. Identify $P(x)$ and $Q(x)$ from the standard form.
2. Compute the integrating factor: $I(x)=e^{\int P(x)dx}$. Do not include a constant of integration here.
3. Multiply the standard form equation by $I(x)$: $$I(x)\frac{dy}{dx}+I(x)P(x)y=I(x)Q(x).$$
4. The left side is now the derivative of a product: $\frac{d}{dx}\left(I(x)\cdot y\right)=I(x)Q(x)$.
5. Integrate both sides with respect to $x$: $$I(x)\cdot y=\int I(x)Q(x)dx+C.$$
6. Finally, solve for $y$ by dividing by $I(x)$.

Example: Solve $x\frac{dy}{dx}-2y=x^3\cos x$.

1. Convert to standard form: $\frac{dy}{dx}-\frac{2}{x}y=x^2\cos x$. So $P(x)=-\frac{2}{x}$, $Q(x)=x^2\cos x$.
2. Integrating factor: $I(x)=e^{\int -\frac{2}{x}dx}=e^{-2\ln |x|}=x^{-2}$.
3. Multiply: $x^{-2}\frac{dy}{dx}-2x^{-3}y=\cos x$.
4. The left side is $\frac{d}{dx}(x^{-2}y)=\cos x$.
5. Integrate: $x^{-2}y=\int \cos xdx=\sin x+C$.
6. Solve: $y=x^2(\sin x+C)$.

Core Method 3: Homogeneous Substitution

Some first-order ODEs are homogeneous in a specific mathematical sense: they can be written in the form $\frac{dy}{dx}=F\left(\frac{y}{x}\right)$. These are not separable in $x$ and $y$, but become separable through a substitution.

The key substitution is $v=\frac{y}{x}$, which implies $y=vx$. Using the product rule, its derivative is $\frac{dy}{dx}=v+x\frac{dv}{dx}$. Substituting these into the original equation transforms it into a separable equation in variables $v$ and $x$.

Process:

1. Confirm the equation is homogeneous (all terms can be written as functions of $\frac{y}{x}$).
2. Let $v=y/x$, so $y=vx$ and $\frac{dy}{dx}=v+x\frac{dv}{dx}$.
3. Substitute to get: $v+x\frac{dv}{dx}=F(v)$.
4. Rearrange to separate: $\frac{1}{F(v)-v}dv=\frac{1}{x}dx$.
5. Solve this separable equation for $v$, then back-substitute $v=y/x$ to express the solution in terms of $y$ and $x$.

Example: Solve $\frac{dy}{dx}=\frac{x^2+y^2}{xy}$.

1. The right side simplifies to $\frac{x}{y}+\frac{y}{x}=\frac{1}{v}+v$, so it's homogeneous.
2. Substitute: $v+x\frac{dv}{dx}=\frac{1}{v}+v\Rightarrow x\frac{dv}{dx}=\frac{1}{v}$.
3. Separate: $vdv=\frac{1}{x}dx$.
4. Integrate: $\frac{1}{2}{v}^2=\ln |x|+C$.
5. Back-substitute: $\frac{1}{2}{\left(\frac{y}{x}\right)}^2=\ln |x|+C$, or $y^2=2x^2(\ln |x|+C)$.

Graphical and Numerical Approaches

Not all differential equations have nice analytical solutions. The IB AA syllabus introduces two powerful alternative approaches.

Slope fields (or direction fields) provide a graphical picture of a differential equation $\frac{dy}{dx}=f(x,y)$. At a grid of points $(x,y)$ on the plane, you draw a short line segment with slope $f(x,y)$. These segments show the "direction" a solution curve would follow if it passed through that point. By sketching curves that are tangent to these segments everywhere, you can visualize families of solution curves without solving the equation analytically.

Euler's method is a simple numerical algorithm for approximating a particular solution. Given an initial point $(x_0,y_0)$ and a step size $h$, it uses the derivative (the slope) to project forward. The recursive formula is: $$x_{n+1}=x_n+h$$ $$y_{n+1}=y_n+h\cdot f(x_n,y_n)$$ While not perfectly accurate, especially with large step sizes, Euler's method demonstrates the fundamental idea of numerical integration and is invaluable for equations that cannot be solved by hand.

Modeling with Differential Equations (HL)

At Higher Level, you are expected to formulate and interpret differential equations in applied contexts. This involves:

1. Translating a verbal description into a differential equation (e.g., "The rate of population growth is proportional to the current population" becomes $\frac{dP}{dt}=kP$).
2. Solving the equation using the appropriate method.
3. Interpreting the solution in the context of the problem, including analyzing long-term behavior (limits as $t\to \infty$) and the meaning of constants.

Common models include exponential growth/decay, limited growth (logistic model), Newton's Law of Cooling, and mixing problems. The emphasis is on setting up the model correctly from the given assumptions.

Common Pitfalls

1. Misapplying the Separable Variables Method: The most frequent error is attempting to separate variables when the equation is not actually separable (e.g., $\frac{dy}{dx}=x+y$). Always check if you can write the equation as a product of a function of $x$ and a function of $y$. If you cannot, another method is required.
2. Forgetting the Constant of Integration and Initial Conditions: The constant $C$ is essential for the general solution. A common sequence mistake is to add $C$ too late or to apply an initial condition before solving for $y$, which can make finding $C$ algebraically messy. It is usually best to solve for $y$ explicitly before substituting initial values.
3. Errors with the Integrating Factor: When computing $I(x)=e^{\int P(x)dx}$, the most common slip is including the $+C$ in the integral. The integrating factor must be a specific function, not a family. Also, after multiplying through by $I(x)$, you must verify the left side collapses to the derivative of $I(x)y$.
4. Neglecting the Back-Substitution in Homogeneous Equations: After solving the separable equation in $v$ and $x$, it is easy to stop and consider $v$ the final answer. You must remember to replace $v$ with $y/x$ to return to the original variables.

Summary

• A differential equation relates a function to its derivatives. Solving one means finding the original function, resulting in a general solution (with $C$) or a particular solution (with an applied initial condition).
• Separable equations are solved by isolating $dy$ and $dx$ on opposite sides and integrating: $\int \frac{1}{g(y)}dy=\int f(x)dx$.
• Linear first-order equations use the integrating factor method, where $I(x)=e^{\int P(x)dx}$ transforms the equation into an easily integrable form.
• Homogeneous equations of the form $\frac{dy}{dx}=F(y/x)$ are solved using the substitution $v=y/x$, turning them into separable equations.
• When analytic solutions are difficult, slope fields give a graphical representation of solution families, and Euler's method provides a step-by-step numerical approximation.
• At HL, the focus shifts to modeling—translating real-world scenarios into differential equations, solving them, and interpreting the results contextually.