Ordinary Differential Equations: One Mark Questions, Book Back Answers and Notes

About Ordinary Differential Equations

Medium ~150 min study

An ordinary algebraic equation relates numbers. A differential equation relates a function to its derivatives, so it describes how a quantity changes. If the rate at which a population grows is proportional to the population, you can write that as $\frac{dP}{dt} = kP$ and solve it to predict the population at any time. The same kind of equation describes the decay of a radioactive substance and the cooling of a cup of tea.

The unit starts with classification. The order of a differential equation is the order of the highest derivative in it, and the degree is the power of that derivative once the equation is free of radicals and fractional powers in the derivatives. You then learn to form a differential equation from a family of curves, such as all circles through the origin with centre on the x-axis, by differentiating and eliminating the arbitrary constants.

Most exam questions ask you to solve a first order equation. Decide the type first: variables separable, homogeneous (substitute $y = vx$), or linear (use the integrating factor). Then solve and apply any initial condition. The application problems on growth, decay, cooling and mixtures all lead to one of these types.

What you'll learn

• Find the order and degree of a differential equation.
• Form the differential equation of a given family of curves.
• Solve equations by the variables separable method and find general and particular solutions.
• Solve homogeneous differential equations using the substitution y = vx.
• Solve first order linear equations using the integrating factor.
• Set up and solve growth, decay, cooling and mixture problems.

Before you start

• Standard integration formulas and methods of integration.
• Differentiation rules, including the product, quotient and chain rules.
• Functions and their graphs, especially exponential and logarithmic functions.

Ordinary Differential Equations explained

First Order Differential Equations: What to Know

Order and Degree

The order is the order of the highest derivative present. The degree is the power of that highest derivative after the equation has been made polynomial in the derivatives. For example, $\left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} = x$ has order 2 and degree 3. If a derivative appears inside a function like $\sin$ or $e^x$, the degree is not defined.

Formation of Differential Equations

Start with the equation of a family of curves containing arbitrary constants. Differentiate as many times as there are constants, then eliminate the constants. A family with $n$ independent arbitrary constants gives a differential equation of order $n$. For instance, $y = mx$ has one constant, and eliminating $m$ gives $x\frac{dy}{dx} = y$.

Variables Separable

If the equation can be written as $f(x)\,dx = g(y)\,dy$, integrate both sides and add one arbitrary constant. That gives the general solution. An initial condition, such as $y = 1$ when $x = 0$, fixes the constant and gives a particular solution.

Homogeneous Equations

An equation $\frac{dy}{dx} = \frac{f(x, y)}{g(x, y)}$ is homogeneous when $f$ and $g$ are homogeneous functions of the same degree, so the right side depends only on $\frac{y}{x}$. Substituting $y = vx$, so that $\frac{dy}{dx} = v + x\frac{dv}{dx}$, turns it into a separable equation in $v$ and $x$. After solving, replace $v$ by $\frac{y}{x}$.

Linear First Order Equations

The standard form is $\frac{dy}{dx} + Py = Q$, where $P$ and $Q$ are functions of $x$ only. The integrating factor is $e^{\int P\,dx}$. Multiplying by it makes the left side the derivative of $y \times \text{I.F.}$, and the solution is $y \times \text{I.F.} = \int Q \times \text{I.F.}\,dx + C$. Remember to put the equation in standard form first, so that the coefficient of $\frac{dy}{dx}$ is 1.

Growth, Decay and Cooling

When a quantity changes at a rate proportional to itself, $\frac{dA}{dt} = kA$, and the solution is $A = A_0 e^{kt}$. For population growth $k > 0$; for radioactive decay $k < 0$. Newton's law of cooling says the rate of change of temperature of a body is proportional to the difference between its temperature and that of the surroundings: $\frac{dT}{dt} = k(T - S)$. It applies to warming as well as cooling.

Mixture Problems

In a mixture problem, a solution flows into a tank while the mixture flows out. If $x$ is the amount of salt in the tank, then $\frac{dx}{dt}$ equals the rate at which salt comes in minus the rate at which it goes out. The outflow rate is the concentration in the tank multiplied by the outflow volume per minute. This gives a separable or linear equation.

Common mistakes to avoid

• Giving the highest power as the order. The order comes from the highest derivative; the degree is its power.
• Finding the degree before removing radicals or fractional powers of the derivatives.
• Leaving out the arbitrary constant, or tacking it on at the end. Add C when you integrate: log y = x + C gives y = Ae^x, not e^x + C.
• Calling an equation linear when it contains terms like y², y times dy/dx, or sin y.
• Finding the integrating factor before writing the equation with coefficient 1 for dy/dx.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

How do I find the degree when there is a radical?

First remove the radical by squaring or cubing so that the derivatives have only whole number powers. Then read off the power of the highest order derivative. For example, if the derivative is under a square root, square both sides before deciding the degree.

Can the degree of a differential equation be zero?

No. The highest derivative is always present, so its power is a positive integer. If the highest derivative appears inside a function such as sin or log and cannot be made polynomial, the degree is not defined.

What is the difference between a general and a particular solution?

A general solution contains arbitrary constants, as many as the order, and represents a family of curves. A particular solution is obtained by using given conditions to find those constants, and it represents one curve from the family.

When should I use the substitution y = vx?

Use it for homogeneous equations, where the numerator and denominator of dy/dx are homogeneous functions of the same degree. The substitution turns the equation into one where the variables v and x can be separated.

Why does the integrating factor work?

Multiplying dy/dx + Py = Q by the integrating factor turns the left side into the derivative of y times the integrating factor, by the product rule. You can then integrate both sides directly.

What does a first order equation describe in applications?

It usually gives the rate of change of one quantity with respect to one variable, often time. Examples are the growth of a population, the decay of a radioactive substance and the temperature of a cooling body.