Forming Differential Equations: Eliminate Arbitrary Constants

You differentiate once, and the constant is still there. You differentiate again, and now there are more terms than you started with. Does that mean you have taken the wrong approach?

Usually, it means you have reached the algebra part of the question.

Forming a differential equation is about finding a rule shared by a whole family of curves. Differentiation supplies information about their slopes and curvature. Elimination then removes the arbitrary constants that distinguish one member of the family from another.

Think of a set of adjustable lamps. Their positions differ, but the mechanism controlling their movement is the same. Your job is to describe that mechanism without naming the setting of each lamp.

What does forming a differential equation mean?

Consider the family:

y = x^2 + C

For C = 1, we get one parabola. For C = 7, we get a higher one. Every member has the same slope at the same value of x:

y' = 2x

The derivative has removed the vertical shift. This first-order differential equation describes the common slope rule.

Here and throughout this guide:

• y' means dy/dx.
• y'' means d^2y/dx^2.
• (y')^2 means the square of the first derivative, which is different from y''.
• e is the exponential constant; ln means natural logarithm.
• Trigonometric arguments are measured in radians.

Forming an equation runs in the opposite direction from solving one. In formation, you start with a family and remove its arbitrary constants. In solving, you start with the derivative relation and try to recover the functions satisfying it.

The basic definitions of differential equations, order, degree, and general solutions are explained in the NCERT chapter on Differential Equations. Our focus here is the elimination working that makes formation questions manageable.

Count independent constants before differentiating

An arbitrary constant can vary from one curve to another, but remains fixed while you differentiate a particular curve with respect to x.

For example, if y = ax^2, then y' = 2ax. The derivative of a is zero, but the derivative of the product ax^2 is not zero.

Also, two letters need not give two independent choices in the curve family.

Given family	Effective arbitrary constants	What to notice
y = ax + b	Two	Slope and intercept vary independently
y = ax + a	One	The same a controls both terms
y = (a + b)x^2	One	Replace a + b by a single C
y = A e^(3x)	One	The number 3 is fixed
y = A e^(kx), with A and k arbitrary	Two	Both amplitude and exponent rate vary

For the usual regular families in school exercises, n independent arbitrary constants lead to an equation of order n. Use this as a planning check, after simplifying the family. It is not a reason to count every letter or to ignore special cases and domain restrictions.

A reliable five-step method

1. Identify the independent variable and the constants the question asks you to eliminate.
2. Simplify the family and count its independent constants.
3. Differentiate sufficiently many times to obtain usable relations.
4. Combine those relations with the original equation to remove the constants.
5. Check the final relation by substituting the given family and its derivatives.

Keep the original equation visible. It is often the very equation you need at the last step.

A useful first question is: can I replace a whole expression containing a constant, instead of solving for that constant separately? This often avoids fractions and unnecessary restrictions.

Example 1: One constant does not mean it vanishes immediately

Form the differential equation of:

y = a(x^2 + 4)

There is one arbitrary constant, a. Differentiate:

y' = 2ax

The constant remains. We now eliminate it using multiplication:

(x^2 + 4)y' = 2ax(x^2 + 4)
2xy = 2ax(x^2 + 4)

Subtract:

(x^2 + 4)y' - 2xy = 0

This is the required first-order equation. We did not divide by x, so the working remains valid at x = 0 too.

Quick check: substitute y = a(x^2 + 4) and y' = 2ax. The two terms cancel exactly, for every a.

If you had differentiated until the polynomial disappeared, you would have reached y''' = 0. That equation is satisfied by every quadratic, including many curves outside the given family. It loses the special link between the constant term and the coefficient of x^2.

Example 2: A constant inside a shifted square

Let:

y = (x + a)^2

Differentiating gives:

y' = 2(x + a)

Square this relation and use the original equation:

(y')^2 = 4(x + a)^2
(y')^2 = 4y

The final equation has order 1 and degree 2. The highest derivative is y', and it appears squared in a polynomial equation in the derivative.

Notice why y'' = 2 would be too broad: it allows y = x^2 + bx + c with two independent constants. Our shifted-square family requires c = b^2/4.

There is a subtle qualification worth knowing. The first-order equation also admits y = 0, which is not an entire shifted-square curve. Algebraic elimination can admit additional solutions. In a formation exercise, first establish that every member of the given family satisfies the result; do not assume that all possible solutions have automatically been classified.

Example 3: Two constants in a quadratic-looking family

Form the equation for:

y = ax^2 + bx

There are two independent constants. Differentiate twice:

y' = 2ax + b
y'' = 2a

Multiply the first derivative equation by x and subtract the original:

xy' - y = ax^2

Since a = y''/2:

xy' - y = x^2y''/2
x^2y'' - 2xy' + 2y = 0

This is second order and first degree. The x^2 coefficient does not make the degree two: degree concerns powers of derivatives, not powers of the independent variable.

Check the cancellation: substituting gives 2ax^2 - 4ax^2 - 2bx + 2ax^2 + 2bx = 0.

The highest-derivative coefficient vanishes at x = 0. The displayed identity still holds there for the original family, but treating it as a standard solved equation for y'' requires an interval with x nonzero.

Example 4: Two exponential terms

Suppose:

y = A e^(2x) + B e^(-3x)

Differentiate twice:

y' = 2A e^(2x) - 3B e^(-3x)
y'' = 4A e^(2x) + 9B e^(-3x)

Look for a combination that cancels both exponential terms. Here:

y'' + y' - 6y
= (4 + 2 - 6)A e^(2x)
  + (9 - 3 - 6)B e^(-3x)
= 0

Therefore:

y'' + y' - 6y = 0

You can derive a reusable pattern. For distinct fixed numbers m and n, the family y = A e^(mx) + B e^(nx) satisfies:

y'' - (m + n)y' + mny = 0

Substitution verifies it because the coefficient of each exponential becomes zero. If m = n, the stated family collapses to (A + B)e^(mx), with only one effective constant. The repeated-rate case needs separate thought rather than automatic use of the two-constant shortcut.

Example 5: A trigonometric family with a fixed frequency

Let:

y = A cos(3x) + B sin(3x)

Then:

y' = -3A sin(3x) + 3B cos(3x)
y'' = -9A cos(3x) - 9B sin(3x)

Recognise the original expression:

y'' = -9y
y'' + 9y = 0

The fixed frequency 3 remains through its square. Forgetting the inner derivative would incorrectly produce y'' + y = 0.

The amplitude-and-phase form y = R sin(3x + alpha) leads to the same equation when R and alpha are arbitrary. Different ways of describing a family can give the same differential equation.

For more practice with those inner factors, see the guide to chain-rule and implicit differentiation.

Example 6: Circles with centres on the x-axis

The wording gives you the family before you differentiate. A circle with centre (a, 0) and variable radius r has equation:

(x - a)^2 + y^2 = r^2

Both a and r are arbitrary. Work on a smooth portion of the circle where y is a differentiable function of x.

Differentiate implicitly and divide by 2:

(x - a) + yy' = 0

Differentiate again, using the product rule on yy':

1 + (y')^2 + yy'' = 0

Both constants have disappeared. This equation has order 2 and degree 1, because the highest derivative y'' occurs to the first power.

The squared first derivative does not change that degree.

At the leftmost and rightmost points of a nondegenerate circle, its tangent is vertical, so a finite derivative dy/dx is not available. That is why the smooth-portion qualification matters. The circle is still perfectly valid geometrically; this choice of dependent variable cannot describe every point with a finite slope.

Example 7: A geometric condition links the constants

Now suppose the circles have their centres on the x-axis and pass through the origin.

Their radius is the distance from (a, 0) to the origin, so the family is:

(x - a)^2 + y^2 = a^2

There is only one independent constant. Expand before differentiating:

x^2 + y^2 = 2ax
2x + 2yy' = 2a

Multiply the derivative relation by x and use the original:

2x^2 + 2xyy' = 2ax = x^2 + y^2

Hence:

x^2 + 2xyy' - y^2 = 0

This is first order. The extra geometric condition has removed one independent choice.

Compare Examples 6 and 7 carefully. The words “pass through the origin” change the constant count, the amount of differentiation needed, and the final order. Writing the correct family is part of the solution, not a preliminary formality.

Example 8: An arbitrary constant in the exponent

Form the equation for:

y = A e^(kx)

This time, both A and k are arbitrary. Differentiating twice gives:

y' = ky
y'' = k^2y

Multiplication eliminates k without division:

yy'' = k^2y^2
(y')^2 = k^2y^2

Therefore:

yy'' - (y')^2 = 0

This method preserves the possibility A = 0, giving y = 0. Taking ln y immediately would exclude zero and negative values of y. Logarithmic differentiation can be useful, but you must keep track of its domain and any omitted cases.

How to choose the next algebra step

What you see	Useful move	Why it helps
One constant multiplying a known function	Multiply original and derivative equations to match terms	Eliminates the constant without unnecessary division
An additive constant	Differentiate	Its derivative is zero
A repeated shifted expression	Express it through y or y’	Avoids expanding everything
A sum of exponentials	Combine y, y’, and y”	Cancels each exponential coefficient
Sines and cosines with fixed frequency	Compare y” with y	Two derivatives restore the original pattern
A circle or another implicit curve	Differentiate implicitly	Preserves the geometry without choosing a square-root branch

These are prompts for thinking, not separate formulas to memorise. In each case the target remains the same: remove the specified arbitrary constants while retaining the family’s useful relation.

Common mistakes and how to repair them

Counting appearances instead of choices. In y = ax + a^2, the same a occurs twice. There is one arbitrary constant. Since y' = a, substitute into the original to get y = xy' + (y')^2.

Dropping a fixed term too early. For y = ax + b + x^3, differentiating twice gives y'' = 6x. The fixed x^3 term is not an arbitrary constant to eliminate.

Forgetting that y depends on x. The derivative of y^2 is 2yy', and the derivative of yy' is (y')^2 + yy''. Writing just 2y or just yy'' breaks implicit examples.

Leaving the last constant behind. If the result still contains a, and a was arbitrary, look back at your original equation before taking another derivative.

Dividing by a quantity that can be zero. From y = Cx^3, use xy' = 3y rather than dividing the original by x^3 without comment. Multiplication often gives a cleaner final relation.

Confusing a necessary relation with an exact solution classification. Substitution confirms that the family satisfies your equation. It does not, by itself, prove that no other solution exists. Squaring, differentiation and singular cases deserve attention if the question asks about the full solution set.

Practice: form the equation, then check your answer

In each question, the capital or lowercase letters described as arbitrary are constants with respect to x.

1. y = x^3 + C, where C is arbitrary.
2. y = Cx^4, where C is arbitrary.
3. y = a(x + 2), where a is arbitrary.
4. y = ax + b + x^3, where a and b are arbitrary.
5. y = A cos(4x) + B sin(4x), where A and B are arbitrary.
6. y = A e^x + B e^(4x), where A and B are arbitrary.
7. y = (a + b)e^(2x), where a and b are arbitrary.
8. y = ax + a^2, where a is arbitrary.

Answers with the key step

Question	Differential equation	Key step
1	y’ = 3x^2	Differentiate once
2	xy’ - 4y = 0	Multiply y’ = 4Cx^3 by x
3	(x + 2)y’ - y = 0	Substitute y’ = a
4	y” - 6x = 0	Two derivatives remove a and b
5	y” + 16y = 0	Apply the inner factor 4 twice
6	y” - 5y’ + 4y = 0	Cancel both exponential terms
7	y’ - 2y = 0	Combine a + b into one constant
8	y - xy’ - (y’)^2 = 0	Replace a by y’ in the original

For question 8, the answer is first order and second degree. For question 6, it is second order and first degree. This pair is a good final check that you are keeping order and degree separate.

A final check before you submit

Read the wording once more. Have you eliminated exactly the constants it calls arbitrary? Is the original geometric condition reflected in your family equation? Did every product and chain rule survive the differentiation?

Then substitute the original family into your final answer. A cancellation that works for arbitrary constants is stronger evidence than checking one convenient numerical example.

If you want to continue from formation to solving, try the guide to separable, homogeneous and linear differential equations.

Further reading

The NCERT Differential Equations chapter provides the foundational vocabulary used here. For additional elimination and curve-family exercises, see Dr. Bandar Al-Mohsin’s King Saud University differential equations notes, particularly the sections titled “The Elimination of Arbitrary Constant” and “Families of Curves.”

Frequently asked questions

What is an arbitrary constant?

It is a value that can differ between members of a family but stays fixed while differentiating a particular member. In y = x^2 + C, changing C selects a different vertical position of the parabola.

How many times should I differentiate?

For a standard regular family, plan around the number of independent arbitrary constants after simplification. One constant usually calls for a first-order relation, two for a second-order relation. Use algebra to eliminate constants rather than differentiating repeatedly until every term disappears.

Must the original equation appear in the working?

Not always, but it is often essential. For y = a(x^2 + 4), the derivative still contains a; combining it with the original removes that constant in one derivative step.

Can a fixed constant remain in the final equation?

Yes. If k is fixed and only A is arbitrary in y = A e^(kx), the required equation is y' - ky = 0. If k is also arbitrary, it must be eliminated too.

Are two letters always two arbitrary constants?

No. In y = (a + b)x^2, the curve depends only on their sum. Writing C = a + b reveals one effective constant. A condition linking two parameters can similarly reduce the number of independent choices.

Why is the derivative of ax not zero when a is constant?

The factor a stays fixed, but x changes. The constant-multiple rule gives d(ax)/dx = a. Only a standalone constant term has derivative zero.

How are order and degree different?

Order identifies the highest derivative present. Degree, when the equation is polynomial in its derivatives, is the power of that highest derivative. Thus (y')^2 = 4y has order 1 and degree 2, while yy'' + (y')^2 + 1 = 0 has order 2 and degree 1.

Can elimination introduce extra solutions?

Yes. For example, (y')^2 = 4y, obtained from shifted parabolas, also admits the identically zero function. Formation establishes a relation satisfied by the family; a complete study of all solutions can require additional analysis.

How do I handle a family described in words?

Write its geometric equation first. Include every restriction before counting constants. Circles with centres on the x-axis have two free parameters, but requiring them to pass through the origin links the radius to the centre and leaves only one.

What is the quickest way to check the final answer?

Differentiate the original family and substitute those expressions into your result. Simplify until both sides match or the residual becomes zero. Also check that no specified arbitrary constant remains and that any division or logarithm respected its domain.