Differential equations

Introduction and mathematical models

A differential equation (DE) is an equation that contains the derivative of an unknown function. It links a quantity with its rate of change. In real life we often know how fast something changes (a population grows by a fixed fraction each year, a hot cup of tea cools faster when it is much hotter than the room). A DE turns that knowledge into a formula for the quantity itself.

DEs are used wherever quantities change continuously in time or space:

• Physics: motion of bodies under forces (Newton's laws) and wave propagation.
• Biology: population growth, the spread of diseases, interaction between species.
• Engineering: electric circuits, control systems and mechanical vibrations.
• Chemistry: reaction rates and diffusion.
• Economics: growth of investments, market behaviour and optimisation.

Mathematical models

A mathematical model is a mathematical description, such as a differential equation, that imitates a real situation so that we can predict how it will behave. Most applications of DEs are mathematical models.

Example: a falling object. Newton's second law says that the net force $F$ on a body equals its mass $m$ times its acceleration. Acceleration is the rate of change of velocity $v$ with time $t$, that is $\frac{dv}{dt}$. So

This law is a differential equation because it contains a derivative. If we ignore air resistance, the only force on an object falling close to Earth's surface is its weight $F=mg$, directed downward, where $g\approx 9.80\mathrm{m}{\mathrm{s}}^{-2}$. The model of the free-falling object is therefore

Translating words into a differential equation

The first SLO of this unit is to construct first order DEs from practical situations. Three phrases do most of the work: **rate of change of $Q$** means $\frac{dQ}{dt}$; is proportional to means "equals a constant $k$ times"; decreases or decays means the rate is negative.

4.1Differential equation

A differential equation is an equation containing one or more derivatives of an unknown function. The book's definition: a mathematical equation containing the derivatives of one dependent variable with respect to one independent variable is called a differential equation (DE).

If the unknown function depends on only one variable, the DE is an ordinary differential equation (ODE). In this unit every DE is ordinary, so we simply call it a DE. Two things must be present: a derivative and an equation (an equals sign).

Dependent variable

The unknown function whose derivatives appear, usually $y$ (or $v$, $P$, $T$ in applications).

Independent variable

The variable with respect to which we differentiate, usually $x$ (or time $t$).

Where a differential equation comes from

Take the function $f(x)=y=x^4+c$, where $c$ is an arbitrary constant (it may have any fixed value). Differentiate with respect to $x$. The constant disappears:

The result $\frac{dy}{dx}=4x^3$ is a differential equation. Every curve $y=x^4+c$, for every value of $c$, satisfies it. So one DE describes a whole family of curves.

The simplest differential equations have the form

where $f$ is a known function of $x$. Some more examples, in each of which $y$ is the dependent variable and $x$ the independent variable:

4.2Order and degree of a differential equationOften examined

Order. The order of a differential equation is the order of the highest order derivative present in the equation.

Types of DE with respect to order

First order DE

A DE whose highest derivative is the first derivative. It can be written $\frac{dy}{dx}=f(x,y)$. Example: $\frac{dy}{dx}=3x$.

Second order DE

A DE whose highest derivative is the second derivative $\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d^{2}y}{d{x}^2}$. Example: $y^{\prime \prime }+x{y}^{\prime }+y=0$.

Degree

Degree. The degree of a differential equation is the power (exponent) of the highest order derivative in the equation, after the equation has been made free of radicals and fractional powers of the derivatives.

Some more types of differential equations are: ordinary DEs (ODE), partial DEs (PDE), linear DEs, non-linear DEs, homogeneous DEs and non-homogeneous DEs. They are described in the next section.

4.2.3Ordinary, linear and non-linear equations

Ordinary differential equation (ODE)

If only one independent variable is involved, the DE is called an ordinary differential equation. It contains one or more derivatives of the unknown function with respect to that single independent variable. The general form of an $n$th order ODE, solved for its highest derivative, is

where $F$ is a known function. This is called an explicit ODE of order $n$. Examples of ODEs:

Linear and non-linear DEs

An $n$th order differential equation is linear if it can be written in the form

that is, it satisfies the following two conditions:

• (a) the dependent variable $y$ and all its derivatives are of power one (and are not multiplied together);
• (b) all the coefficients $a_n(x),a_{n-1}(x),\dots ,a_1(x),a_0(x)$ and the function $f(x)$ are constants or depend only on the independent variable $x$.

If any one of these two conditions is not satisfied, the DE is non-linear. Signs of non-linearity: a power of $y$ or of a derivative ($y^2$, $(y^{\prime }{)}^3$), a product such as $y{y}^{\prime \prime }$, or a function of $y$ such as $\sin y$ or $e^y$.

4.3Concept of solution of a differential equation

Differential equations are studied mainly through their solutions: the functions that satisfy them.

A solution of a differential equation is any function y=f(x), defined on some interval I, that reduces the equation to an identity (a statement true for every x in I) when y and its derivatives are substituted.

To solve a DE such as \dfrac{dy}{dx}-x=0 means to find the unknown function, either explicitly as y=f(x) or implicitly as a relation between x and y.

The simplest case: integrate

Consider the first order DE \dfrac{dy}{dx}=f(x). If f(x) is continuous, integrating both sides gives

where F(x) is an anti-derivative of f(x) and c is the constant of integration.

This example shows that a DE has two types of solutions: the general solution and a particular solution.

4.3.1General and particular solutionsOften examined

General solution

The solution that contains as many arbitrary constants as the order of the differential equation. It is a relation between the independent variable x and the dependent variable y, obtained by removing the derivatives (by integration). In Example 3, x^{2}+y^{2}=c is the general solution.

Particular solution

A solution free from arbitrary constants. It is obtained by giving particular values to the arbitrary constants. In Example 3, x^{2}+y^{2}=4 and x^{2}+y^{2}=9 are particular solutions.

Trivial solution

A solution of a DE that is identically zero on an interval, such as y=0 in Examples 4 and 5.

4.4Formation of differential equationsOften examined

We can form a differential equation by eliminating the arbitrary constants from an algebraic equation (which is then a solution of the DE). Each differentiation gives one more equation, so to remove n constants we differentiate n times.

4.4.1Explicit and implicit solutions

Explicit solution

A solution that can be written in the form y=f(x), with y alone on one side.

Implicit solution

A solution in the form of a relation f(x,y)=0, not solved for y.

4.4.2Number of solutions

A given differential equation usually has an infinite number of solutions, one for each value of the arbitrary constant(s).

4.5Solution of first order, first degree equations

The solution of an algebraic equation such as x^{2}-5x+6=0 is a number: the values x=2 and x=3 satisfy it. The solution of a differential equation such as y\,dy=x\,dx is a function. To find it we remove the derivative by some technique, and every technique involves integration. After removing the derivative, the dependent variable y is given in terms of the independent variable x, either explicitly (y=f(x)) or implicitly (g(x,y)=0).

A DE of first order and first degree can be written as \dfrac{dy}{dx}=f(x,y) or as M(x,y)\,dx+N(x,y)\,dy=0. Some standard forms have ready methods:

• variable separable differential equations,
• equations reducible to variable separable form,
• homogeneous differential equations,
• equations reducible to homogeneous form.

4.5.1Variable separable equationsOften examined

A first order differential equation of the form

where f is a function of x only and g is a function of y only, is called separable, or is said to have separable variables.

For example, \dfrac{dy}{dx}=y^{3}xe^{2x+y} is separable, because it can be written as \dfrac{dy}{dx}=\left(xe^{2x}\right)\left(y^{3}e^{y}\right)=f(x)\,g(y). But \dfrac{dy}{dx}=y+\sin x is not separable: y+\sin x cannot be written as a product f(x)\,g(y).

4.5.1Initial condition and initial value problemOften examined

The general solution of a DE contains as many arbitrary constants as the order of the DE. Often we need the one solution that satisfies an extra (supplementary) condition, such as a known value of y at a given x.

Suppose we want the solution of \dfrac{dy}{dx}=f(x,y) with y=y_0 at x=x_0. Substituting x=x_0 and y=y_0 in the general solution gives a particular value of the constant, and so a particular solution. The condition y(x_0)=y_0 is called the initial condition, and the DE together with it is an initial value problem (IVP):

4.6.1Homogeneous functions

Before homogeneous differential equations we need homogeneous functions. Roughly, a polynomial in x and y is homogeneous when every term has the same total degree (power of x plus power of y).

Definition. If a function f has the property

where t\in\mathbb{R}^{+} and n\in\mathbb{R}, then f is a **homogeneous function of degree n**. To test a function, replace x by tx and y by ty, simplify, and see whether a power t^{n} comes out as a common factor.

4.6.2Homogeneous differential equationsOften examined

A differential equation of the form

is homogeneous if M and N are homogeneous functions of the same degree n, that is M(tx,ty)=t^{n}M(x,y) and N(tx,ty)=t^{n}N(x,y).

Equivalently, the equation \dfrac{dy}{dx}=f(x,y) is homogeneous when f is homogeneous of degree 0; then f depends only on the ratio \dfrac{y}{x}. Such an equation is reduced to a separable one by substituting y=ux (or x=vy), where u (or v) is a new dependent variable.

4.7Applications of differential equationsOften examined

Differential equations are used to solve problems in engineering and the physical sciences, such as physics, chemistry, biology and economics. First order ODEs describe the flow of electricity, the motion of a pendulum and of a falling object, heat flow (thermodynamics) and population growth. In medicine they are used to follow the growth of diseases through graphs.

4.7Population growth and decayOften examined

In 1798 Thomas Malthus proposed that the rate of increase of the population of a country is proportional to the population at that time. If P(t) is the population at time t,

where k is the constant of proportionality, called the growth constant. k>0 gives growth and k<0 gives decay. The same DE describes bacteria in a culture, money at continuous compound interest and radioactive decay.

**Finding k from two readings.** If P=P_1 at t=t_1, then P_1=P_0e^{kt_1}, so

Doubling time. The population doubles when e^{kt}=2, that is after t_d=\dfrac{\ln2}{k}, whatever the starting value.

4.7Radioactive decay and half-life

Radioactive substances disintegrate at a rate proportional to the amount present. If A(t) is the amount (mass or number of atoms) at time t,

The minus sign shows that A decreases, and \lambda is the decay constant. Solving exactly as in the growth equation (with k=-\lambda) gives A=A_0e^{-\lambda t}, where A_0 is the amount at t=0.

4.7Newton's law of cooling and warmingOften examined

Newton's law of cooling: the rate at which a body cools is proportional to the difference between its temperature T and the temperature T_s of the surrounding medium (air, water, ice). The same law describes the warming of a cold body placed in a warm room.

Here t is time and \dfrac{dT}{dt} is the rate of change of the temperature. For a cooling body T>T_s and T decreases, so \dfrac{dT}{dt}<0: the constant k is negative. (Many books write \dfrac{dT}{dt}=-k(T-T_s) with k>0; both forms give the same answers.)

4.7Flow of electricity: series circuits

The SLOs of this unit also mention the flow of electricity in a series circuit. The book gives no worked example, so here is the standard model. It is solved by separating the variables.

By Kirchhoff's voltage law, the sum of the voltage drops round the loop equals the applied e.m.f. E. The drop across the resistance R is Ri, and across the inductance L it is L\dfrac{di}{dt}, where i(t) is the current. So

RC circuit. A capacitor of capacitance C with charge q discharging through a resistance R obeys Ri+\dfrac{q}{C}=0 with i=\dfrac{dq}{dt}. So \dfrac{dq}{dt}=-\dfrac{q}{RC}, whose solution is q=q_0e^{-t/(RC)}: exponential decay, exactly like radioactive decay.

4.7Economics and finance

When interest is added continuously, the amount A in an account grows at a rate proportional to itself. If r is the annual interest rate (as a decimal) and t is the time in years,

where A_0 is the amount invested. This is the growth equation again. Similarly, a machine whose value V falls at a rate proportional to its value (depreciation) obeys \dfrac{dV}{dt}=-kV, so V=V_0e^{-kt}.

Summary: choosing the method

Formula sheet

Key terms

Differential equation (DE)

An equation containing one or more derivatives of an unknown function.

Ordinary differential equation (ODE)

A DE in which the unknown function depends on only one independent variable, so only ordinary derivatives appear.

Partial differential equation (PDE)

A DE whose unknown depends on two or more independent variables, so it contains partial derivatives.

Dependent variable

The unknown function whose derivatives appear in the DE, usually y.

Independent variable

The variable with respect to which we differentiate, usually x or t.

Order

The order of the highest order derivative present in the DE.

Degree

The power of the highest order derivative, after the DE is made free of radicals and fractional powers of derivatives. It is not defined if a derivative is inside a function such as \log or \sin.

Linear DE

A DE in which y and all its derivatives have power 1 and every coefficient depends only on x (or is constant).

Non-linear DE

A DE that is not linear, for example one containing y^{2}, (y')^{3}, y\,y'' or \sin y.

Solution of a DE

A function, defined on an interval, that turns the DE into an identity when substituted.

Arbitrary constant

A constant that may take any value, such as the constant of integration c.

General solution

A solution containing as many arbitrary constants as the order of the DE; it represents a family of curves.

Particular solution

A solution with no arbitrary constants, obtained by giving the constants particular values; it is one curve of the family.

Trivial solution

A solution that is identically zero on an interval, such as y=0.

Family of curves

The set of curves obtained from a general solution by giving its constant different values.

Explicit solution

A solution written as y=f(x).

Implicit solution

A solution given as a relation f(x,y)=0 that is not solved for y.

Initial condition

A condition y(x_0)=y_0 that fixes the value of the arbitrary constant.

Initial value problem (IVP)

A DE together with an initial condition, \dfrac{dy}{dx}=f(x,y), y(x_0)=y_0.

Separable DE

A first order DE that can be written \dfrac{dy}{dx}=f(x)\,g(y), so the variables can be separated and integrated.

Homogeneous function

A function with f(tx,ty)=t^{n}f(x,y) for all t>0; n is its degree.

Homogeneous DE

A DE M\,dx+N\,dy=0 in which M and N are homogeneous functions of the same degree; it is solved by y=ux.

Slope field

A picture of short line segments showing the slope \dfrac{dy}{dx} given by a DE at many points; solution curves follow the segments.

Mathematical model

A mathematical description, such as a DE, that imitates a real situation so that its behaviour can be predicted.

Malthusian law

The rate of increase of a population is proportional to the population: \dfrac{dP}{dt}=kP.

Growth constant

The constant k in \dfrac{dP}{dt}=kP; positive for growth, negative for decay.

Decay constant

The constant \lambda>0 in \dfrac{dA}{dt}=-\lambda A.

Half-life

The time in which half of a radioactive substance decays: T_{1/2}=\dfrac{\ln2}{\lambda}.

Newton's law of cooling

The rate of change of a body's temperature is proportional to the difference between its temperature and that of its surroundings.

Superposition principle

For a linear DE with right side 0, constant multiples and sums of solutions are again solutions.

FBISE 2026 questions

Questions from the FBISE HSSC-II 2026 (1st Annual) papers that belong to this unit, answered the way an examiner expects.

Exam focus

1. Find the order and degree of a DE, including one with a derivative under a root such as \dfrac{dy}{dx}=\sqrt{1+\left(\dfrac{d^{2}y}{dx^{2}}\right)^{2}}. (short)
   A regular MCQ and short question; the 2026 Old syllabus paper asked for the order in MCQ 16.
   Go to answer
2. Form the DE by eliminating the arbitrary constants from y=a\cos x+b\sin x or y=bx^{2}+2ax. (problem)
   Tests differentiation and elimination; Examples 6 and Exercise 4.1 Q2.
   Go to answer
3. Verify that a given function is a solution of a DE, for example y=5\tan5x for y'=25+y^{2}. (problem)
   Quick marks by substitution; Exercise 4.1 Q3 and Review Q3.
   Go to answer
4. Solve a separable DE such as \dfrac{dy}{dx}=\dfrac{y+1}{x-2}. (problem)
   Asked for 4 marks in both 2026 new-syllabus sets (Q.2 (iv)) and in both old-syllabus papers (Q.2 (iv) OR, Q.2 (xi)).
   Go to answer
5. Solve an initial value problem such as \dfrac{dy}{dx}=y^{2}+4, y(0)=-2. (problem)
   Combines separation with finding the constant and choosing the correct branch; Exercise 4.2 Q9 to Q15.
   Go to answer
6. Test whether f(x,y) is homogeneous and find its degree. (short)
   Exercise 4.3 Q1 to Q3, Example 13 and Review MCQ (iv).
   Go to answer
7. Solve a homogeneous DE by y=ux, for example \dfrac{dy}{dx}=\dfrac{x^{2}+y^{2}}{xy}. (problem)
   The 8 mark Section C question in 2026 Set B, Set D and the Old syllabus paper.
   Go to answer
8. Population growth: find k from two readings and predict a future value. (problem)
   2026 Sets B and D Q.2 (i) OR; Exercise 4.4 Q1 and Q3; Set D MCQ 9.
   Go to answer
9. Newton's law of cooling: find the temperature as a function of time, or the time to reach a given temperature. (problem)
   Example 17 and Exercise 4.4 Q2 and Q5.
   Go to answer
10. Radioactive decay: write the DE and use the half-life. (problem)
    Exercise 4.4 Q4; the 2026 Old syllabus MCQ 17 asked for \dfrac{dM}{dt}=-kM.
    Go to answer
11. Define general and particular solutions and state how many arbitrary constants a general solution has. (short)
    Definitions for short questions and MCQs (Review MCQ (vi)).
    Go to answer