A differential equation is simply an equation that contains a derivative — it links a function to its rate of change. That sounds heavy, but for NDA it is one of the most mechanical and predictable chapters: spot the type, apply the matching method, and the answer falls out. It quietly carries 4–6 marks every year, and the question patterns barely change.
Why Differential Equations Are Easy Marks
Calculus is the backbone of NDA Maths, and differential equations sit right at the top of that backbone. Many students fear the name, but once you see how few methods are actually tested, the fear disappears. The whole chapter runs on a tiny toolkit: separate the variables, spot a homogeneous form, or build an integrating factor. Choose the right one and the rest is routine algebra.
What makes this chapter a Cavalier favourite is its predictability. The examiner almost never invents anything exotic. Year after year the questions ask you to find the order and degree, form an equation by eliminating constants, or solve a first-order equation by one of three standard methods. There is rarely a surprise waiting to eat your time.
Because the methods are so fixed, this is a chapter you can almost guarantee. Unlike a tricky geometry figure that can stall you for ten minutes, a differential equation rewards pattern recognition over cleverness.
Before you try to solve any differential equation, first classify it. Ask: is it variable separable, homogeneous, or linear? Naming the type instantly tells you which method to use and saves you from blind algebra.
What a Differential Equation Actually Is
A differential equation (DE) is any equation involving an unknown function and one or more of its derivatives. The unknown is no longer a number like in x + 3 = 7; it is a whole function y = f(x) that we must rebuild from information about its slope.
Examples of differential equations:
dy/dx = 2x
d2y/dx2 + y = 0
(dy/dx)2 − 3y = x
The solution of a DE is the function that satisfies it. A general solution carries arbitrary constants (one for each order), while a particular solution fixes those constants using given conditions. So solving a DE is really integration in disguise — you are undoing the derivatives.
For NDA you mainly meet ordinary differential equations — equations with derivatives of a single independent variable, usually x. Equations with partial derivatives (∂) are not part of the core scoring set.
Order and Degree: The First Two Marks
Almost every year the paper hands you free marks by asking for the order and degree of a given equation. Get the definitions exact and these questions take ten seconds.
Order = the order of the highest derivative present.
Degree = the power of that highest-order derivative, after the equation is made free of radicals and fractions in the derivatives, and is a polynomial in the derivatives.
So in d2y/dx2 + (dy/dx)3 + y = 0, the highest derivative is the second derivative, so the order is 2. Its power is 1, so the degree is 1. The cube on dy/dx does not matter, because degree only counts the power of the highest-order derivative.
Degree is only defined when the equation can be written as a polynomial in the derivatives. If a derivative sits inside sin, log, or under a root that cannot be cleared, the degree is not defined. Students wrongly write 1 instead of "not defined".
If you see a square root or fractional power on a derivative, first square or rationalise to clear it, then read off the degree. The equation must be polynomial in derivatives before you count.
Formation of a Differential Equation
Formation is the reverse skill: you are given a family of curves with arbitrary constants, and you must produce a differential equation that has no constants left in it. The rule is beautifully simple.
To form a DE, differentiate the given relation as many times as there are arbitrary constants, then eliminate those constants between the equations.
n arbitrary constants → differentiate n times → a DE of order n.
So a family with one constant gives a first-order equation, and a family with two constants gives a second-order one. For example, take y = mx, a family of straight lines through the origin with one constant m. Differentiating once: dy/dx = m. Substitute m back: dy/dx = y/x. The constant has vanished, and we have a first-order DE for the whole family.
Form the DE of the family y = A·ex, where A is an arbitrary constant.
The order of the formed DE equals the number of arbitrary constants in the original family. Count the constants first and you already know the order of your answer.
Method 1: Variable Separable
This is the gentlest and most common first-order method. If you can shuffle the equation so that all the y terms sit with dy and all the x terms sit with dx, you simply integrate both sides.
If dy/dx = f(x)·g(y), rewrite as dy/g(y) = f(x) dx, then integrate:
∫ dy/g(y) = ∫ f(x) dx + C
The whole trick is the separation. Once x lives on one side and y on the other, the problem is just two ordinary integrals. The single constant C is added on one side only — you never need two.
Solve dy/dx = xy.
Do not forget the constant of integration. In a general solution it must appear, and how you write it — as C, log C, or k = eC — can change how neat the final answer looks. Choose the form that simplifies the algebra.
Method 2: Homogeneous Differential Equations
An equation dy/dx = F(x, y) is homogeneous when the right side can be written purely as a function of the ratio y/x. In other words, every term has the same total degree in x and y.
If dy/dx = g(y/x), substitute y = vx, so that dy/dx = v + x·dv/dx.
This converts the equation into a variable separable one in v and x.
The substitution is the entire skill here. After putting y = vx, the equation always becomes separable, and you finish with the method you already know. At the very end you replace v with y/x to return to the original variables.
To test for homogeneity quickly, replace x with λx and y with λy. If every term scales by the same power of λ and that power cancels, the equation is homogeneous and the y = vx substitution will work.
After substituting y = vx, students often forget that dy/dx becomes v + x·dv/dx, not just dv/dx. Missing the extra v term wrecks the separation and the final answer.
Method 3: Linear DEs and the Integrating Factor
This is the most powerful first-order method and a perennial NDA favourite. A linear differential equation has the standard form below, where P and Q depend only on x.
Standard form: dy/dx + P·y = Q
Integrating Factor: I.F. = e∫ P dx
Solution: y · (I.F.) = ∫ Q · (I.F.) dx + C
The integrating factor is the magic ingredient. Multiplying the whole equation by it turns the left side into the neat derivative of (y × I.F.), so integrating becomes trivial. The recipe never changes: write in standard form, find P, compute the I.F., then apply the solution formula.
Solve dy/dx + y = x, a linear equation with P = 1 and Q = x.
The equation must be in the exact form dy/dx + P·y = Q before you read off P. If the coefficient of dy/dx is not 1, divide through first to make it 1, otherwise your P is wrong.
Worked Example: Putting It All Together
Let us solve a homogeneous equation from start to finish, since it combines two methods in one and is exactly the multi-step flavour NDA likes.
Solve dy/dx = (x + y) / x.
Notice the flow: we first recognised the homogeneous form, used the y = vx substitution, and then finished with simple variable separation. The v term cancelled neatly, which is what usually happens when the equation is genuinely homogeneous — a good sign that you set it up correctly.
Always do a quick sanity check at the end. Differentiate your answer mentally and see if it could regenerate the original equation. A thirty-second check catches sign slips before they cost a mark.
Common Mistakes That Cost Easy Marks
Most errors here are mechanical, not conceptual. Knowing the usual traps protects almost all of your marks.
- Confusing order and degree — order is about the highest derivative, degree is its power after clearing radicals.
- Calling degree 1 when a derivative sits inside sin, log or a stubborn root, where degree is actually not defined.
- Dropping the constant of integration — a general solution must carry C.
- Forgetting dy/dx = v + x dv/dx in homogeneous problems.
- Reading P wrongly in a linear DE because the equation was not in standard form first.
When counting arbitrary constants during formation, count independent constants only. Two constants that can be merged into one (like A + B written as a single C) give a lower order than students expect.
Previous-Year Style Question
Here is a question in the exact flavour the NDA exam loves — a short classification problem that tests whether your definitions are crisp.
Q. What is the order and degree of the differential equation [1 + (dy/dx)2]3/2 = k · d2y/dx2?
Answer: The highest derivative is the second derivative, so the order is 2. To find the degree, the equation must be a polynomial in derivatives, so square both sides to clear the 3/2 power: [1 + (dy/dx)2]3 = k2 (d2y/dx2)2. Now the highest-order derivative appears to the power 2, so the degree is 2. Order 2, degree 2.
Notice that the cube of dy/dx played no role in the degree — only the power of the highest-order derivative counts. Clearing the fractional power first is the whole trick the examiner is checking.
Quick Revision Before the Exam
Run through this checklist the night before and differential equation questions will feel routine.
- Order = highest derivative present; Degree = its power after clearing radicals (else not defined).
- Formation: differentiate once per arbitrary constant, then eliminate the constants.
- n arbitrary constants give a DE of order n.
- Variable separable: get dy/g(y) = f(x) dx, then integrate both sides.
- Homogeneous: write as g(y/x), put y = vx with dy/dx = v + x dv/dx.
- Linear: dy/dx + Py = Q; I.F. = e∫P dx; y·(I.F.) = ∫ Q·(I.F.) dx + C.
- Always add the constant of integration in a general solution.
Practise five mixed differential equations daily for a fortnight, and this chapter becomes one of your most dependable scorers. At Cavalier, students who drill these three methods rarely lose a single differential-equation mark in the actual NDA exam.
Frequently asked questions
What is the difference between the order and degree of a differential equation?
The order is the order of the highest derivative that appears in the equation, while the degree is the power of that highest-order derivative after the equation has been made free of radicals and fractions in the derivatives. Degree is only defined when the equation is a polynomial in its derivatives.
How many marks do differential equations carry in the NDA exam?
Differential equations typically account for about 4 to 6 marks in each NDA Maths paper. Because the question patterns repeat closely, focusing on order, degree, formation, and the three first-order methods makes it a reliable scoring area.
How do I decide which method to use to solve a differential equation?
First classify the equation. If you can separate x and y terms, use variable separable. If the right side is a function of y/x, it is homogeneous and you substitute y equals vx. If it has the form dy/dx plus Py equals Q, it is linear and you use the integrating factor.
What is an integrating factor and when is it used?
An integrating factor is e raised to the integral of P with respect to x, used for linear differential equations of the form dy/dx plus Py equals Q. Multiplying the equation by it turns the left side into the derivative of y times the integrating factor, making the equation easy to integrate.
How is the order of a differential equation related to the number of arbitrary constants?
When you form a differential equation by eliminating arbitrary constants from a family of curves, the order of the resulting equation equals the number of independent arbitrary constants. So a family with one constant gives a first-order equation and one with two constants gives a second-order equation.
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