Differential Equations

This article explores differential EquationsWe will first look at the definition of a differential equation and how to verify a solution. We then look at how to solve a separable first order ordinary differential equation, sketch families of solutions, and model with a differential equation. 

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Table of contents

    What is a differential equation?

    When we have an equation that involves a series of derivatives, we call this a differential equation. When the derivatives are of a function of one variable, we call this an ordinary differential equation (ODE). When talking about a differential equation, we often talk about its order. Order means the highest derivative that is present in the equation. For example, the equation is of order two, as the highest order derivative in the equation is second order.

    When solving a differential equation, the aim is to find a function that satisfies the equation. This solution will not be unique, as with a derivative, a constant can be added to change the function but still satisfy the equation. The only way that the value of this constant can be found is by the addition of a boundary condition.

    In a first order ordinary differential equation, we only need one boundary condition to satisfy the unknown. In general, for an nth order ordinary differential equation, we need n boundary conditions. A boundary condition specifies the value of the function at a certain point. This allows you to then work out the value of any unknown coefficients.

    Verifying solutions of differential equations

    When given a differential equation, if we get given a potential solution, we can verify whether it is valid or not. This involves working out all the derivatives used and then filling these in to see if the potential solution is suitable to satisfy the equation.

    Verify that is a solution to . (Note here that we use yιto represent , and yιι to represent ).

    Using the Product Rule, let us find the first and second derivative of y with respect to x.



    We can now fill in the values to get

    Hence the solution is verified.

    Solving differential equations

    At A level, we only need to know how to solve first order separable ordinary differential Equations. Separable refers to the fact that the two variables (normally x and y) can be separated and then split up to solve.

    The form of a separable differential equation (for variables y and x, where y is a function of x) is . We can then rearrange this to, and then integrate to get. Once integrated, this is our general solution for the differential equation. We can then apply any boundary conditions to find a specific solution if necessary.

    It is worth noting that strictly speaking, we cannot manipulate in this way, as it is not a fraction but rather a Notation for the derivative. However, we can treat it like a fraction in this case.

    Find the general solution to.

    This is of the form , so that means that the solution is of the form . This means we can fill in the two Functions to get to . Integrating the right-hand side, we get . On the left-hand side, this is a standard integral, given as .

    This can also be achieved by using the substitution

    This means that our solution is given as . Note we have combined both the constants into one here. We can simplify this further to give.

    Find the solution to , with.

    First, let us separate this equation to get .

    The left-hand side integrates to , and the right-hand side integrates to.

    These combine to give. This can simplify further to give.

    We know that, so we can fill this in to get.

    This gives C = 1, and our solution is.

    Sketching a family of solution curves for differential equations

    When we are finding a general solution for a differential equation, then we are left with constants in the general equation. These constants can be any value, and they would still satisfy the differential equation. The family of solution curves is the collection of the Functions with various values for the constants.

    Find the general solution to , and draw a graph showing this with four different particular solutions.

    Below is a graph showing when C = -1, 0, 1, 2

    Differential Equations A family of solutions StudySmarterA family of solutions, with C = 2 green, C = 1 blue, C = 0 red and C = -1 purple, Tom Maloy - StudySmarter Originals

    Modeling with differential equations

    The reason we study differential equations is the ability to use them in real-life scenarios. To illustrate this, let's look at an example.

    Suppose there is a cylindrical water tank of radius 5m. The height of water in the tank at any point is denoted as. Water flows out of the tank at a rate proportional to the square root of the volume in the tank. Find .

    The volume in the water tank at any one point is given as, meaning.

    The water flows out at a rate proportional to the square root of the volume, meaning, where a is a constant of proportionality.

    We can use the earlier expression for volume to give.

    Then, by the Chain Rule,.

    Differential Equations - Key takeaways

    • A differential equation is an equation made up of various derivatives.
    • The order of a differential equation is the highest order of any derivative in the equation.
    • A separable first-order ordinary differential equation has the form , with solution.
    • Boundary conditions allow us to put a value on a constant.
    • A general solution is one that has an unknown constant in it, and a specific solution exists for a specific boundary condition.
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    Frequently Asked Questions about Differential Equations

    What is a differential equation?

    A differential equation is an equation involving derivatives of a function, and the solution is finding that solution.

    How do you solve differential equations?

    At A-Level, we need only solve a first order separable ordinary differential equation of the form dy/dx=f(x)g(y), which has a solution of ∫ 1/g(y )dy=∫ f(x)dx

    What does a differential equation represent?

    A differential equation can represent pretty much any system. Anything which changes with time, position, etc. can be represented by differential equations.

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