Mathematics Class 12 — Part…
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Differential Equations — Introduction, Order and Degree

🎓 Class 12 Mathematics CBSE Theory Ch 9 — Differential Equations ⏱ ~15 min
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This MCQ module is based on: Differential Equations — Introduction, Order and Degree

This mathematics assessment will be based on: Differential Equations — Introduction, Order and Degree
Targeting Class 12 level in General Mathematics, with Advanced difficulty.

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9.1 Introduction

In Chapter 7 (Integrals) you computed antiderivatives — given \(f(x)\), find \(F(x)\) with \(F'(x)=f(x)\). That problem is the simplest possible differential equation?: \(\dfrac{dy}{dx}=f(x)\). In real applications, however, the relationship between the unknown function and its derivatives is often more involved — for instance,

\[\dfrac{dy}{dx}=g(x,y),\qquad \dfrac{d^2y}{dt^2}+\omega^2 y=0,\qquad \dfrac{dN}{dt}=kN.\]

The first models any process where the rate of change depends on both position and time; the second is the simple harmonic oscillator (springs, pendulums); the third is exponential growth/decay (population, radioactive decay, compound interest). Differential equations are the language of every dynamical process in nature.

Henri Poincaré

1854 – 1912

French polymath — mathematician, theoretical physicist, and philosopher. Often called "the last universalist", Poincaré transformed the qualitative theory of differential equations: rather than seeking explicit formulas, he studied the geometry of solution curves (phase portraits, stability, periodic orbits). His work founded chaos theory and modern dynamical systems.

9.2 Basic Concepts

Definition: Differential Equation
A differential equation is an equation involving the derivatives of a dependent variable \(y\) with respect to one or more independent variables. If only one independent variable is involved, it is called an ordinary differential equation (ODE); otherwise partial (PDE). This chapter studies ODEs only.

Examples: \(\dfrac{dy}{dx}=e^x\), \(y\dfrac{dy}{dx}=x\), \(x^2\dfrac{d^2y}{dx^2}+y=0\).

9.2.1 Order

Order
The order of a differential equation is the order of the highest-order derivative occurring in it.
DEOrder
\(\dfrac{dy}{dx}+y=x\)1
\(\dfrac{d^2y}{dx^2}+5\dfrac{dy}{dx}+6y=0\)2
\(\dfrac{d^3y}{dx^3}+\!\!\left(\dfrac{dy}{dx}\right)^4=\sin x\)3

9.2.2 Degree

Degree
After clearing all radicals and fractional powers (so the equation is polynomial in the derivatives), the degree is the highest power of the highest-order derivative.
If the equation cannot be reduced to a polynomial in the derivatives, the degree is undefined.

Examples:

  • \(\dfrac{d^2y}{dx^2}+5x\!\!\left(\dfrac{dy}{dx}\right)^2-6y=\log x\) — order 2, degree 1 (the \(d^2y/dx^2\) appears to the first power).
  • \(4\dfrac{d^4y}{dx^4}\sin\!\!\left(\dfrac{dy}{dx}\right)=0\) — degree undefined (sin of a derivative is not a polynomial).
  • \(\left(\dfrac{dy}{dx}\right)^3-3\dfrac{dy}{dx}+y=0\) — order 1, degree 3.

9.3 General and Particular Solutions

Solution of a DE
A function \(y=\phi(x)\) is a solution of a differential equation if substituting it into the equation makes the equation an identity. The graph of a solution is called a solution curve or integral curve.

Example: \(y=A\cos x+B\sin x\) is a solution of \(\dfrac{d^2y}{dx^2}+y=0\) for any constants \(A,B\). Substituting: \(y''=-A\cos x-B\sin x=-y\), so \(y''+y=0\). ✓

General vs Particular solution
The general solution of an \(n\)-th order DE contains \(n\) arbitrary constants. A particular solution is obtained by giving specific values to those constants — typically using initial conditions like \(y(x_0)=y_0\), \(y'(x_0)=y_1\).

9.3.1 Formation of a Differential Equation from a Family of Curves

Starting from a family of curves \(F(x,y,c_1,c_2,\ldots,c_n)=0\) (with \(n\) arbitrary constants), differentiate \(n\) times and eliminate the constants. The result is an \(n\)-th order DE that the family satisfies.

Worked Examples

Example 1. Find the order and degree of \(\dfrac{d^4y}{dx^4}-\sin\!\!\left(\dfrac{d^3y}{dx^3}\right)=0\).
Order: 4 (highest derivative \(d^4y/dx^4\)). Degree: not defined — the equation contains \(\sin(d^3y/dx^3)\) which is not a polynomial in derivatives.
Example 2. Show that \(y=ax+b/x\) is a solution of \(x^2 y''+xy'-y=0\).
\(y'=a-b/x^2\), \(y''=2b/x^3\). Substitute: \(x^2(2b/x^3)+x(a-b/x^2)-(ax+b/x)=2b/x+ax-b/x-ax-b/x=0\). ✓
Example 3. Verify that \(y=Ae^{2x}+Be^{-x}\) is the general solution of \(y''-y'-2y=0\).
\(y'=2Ae^{2x}-Be^{-x}\); \(y''=4Ae^{2x}+Be^{-x}\). \(y''-y'-2y=(4A-2A-2A)e^{2x}+(B+B-2B)e^{-x}=0\). ✓ Two arbitrary constants A, B match the order 2.
Example 4. Form the DE of the family \(y=mx\) (lines through the origin), where \(m\) is the parameter.
One parameter \(m\), so order 1 DE expected. Differentiate: \(y'=m\). From \(y=mx\), \(m=y/x\). So \(y'=y/x\), i.e. \(\dfrac{dy}{dx}=\dfrac{y}{x}\).
Example 5. Form the DE for circles of radius 5 centred on the x-axis: \((x-h)^2+y^2=25\).
One parameter \(h\). Differentiate: \(2(x-h)+2yy'=0\Rightarrow x-h=-yy'\). Substitute back: \((yy')^2+y^2=25\), i.e. \(y^2(1+(y')^2)=25\). Order 1, degree 2.
Activity: Order and Degree Identification
L3 Apply
Materials: Pen, paper.
Predict: What's the order and degree of \(\sqrt{1+(y')^2}=k\,y''\)?
  1. Square both sides to remove the radical: \(1+(y')^2=k^2(y'')^2\).
  2. Highest-order derivative is \(y''\), so order = 2. Highest power of \(y''\) is 2, so degree = 2.
  3. Now try \(\dfrac{dy}{dx}=e^{y''}\). Order = 2 (because of \(y''\)). Degree = undefined (exponential of a derivative isn't polynomial).
  4. Try \((y''')^2+(y'')^4+y'+y=x\). Order = 3, degree = 2.
  5. Lesson: order is easy (just look for the highest derivative); degree requires the equation to be polynomial in derivatives.
Order is the structural complexity (highest derivative); degree quantifies the leading nonlinearity. Linear DEs (degree 1, no products of derivatives) have powerful theory; nonlinear DEs are much harder. Most physics is linear or "linearisable" precisely to leverage this.

Competency-Based Questions

Scenario: A balloon being inflated has volume V(t) growing such that dV/dt = k·V — i.e. the rate of volume increase is proportional to current volume. Initial volume V(0)=V₀.
Q1. Order and degree of \(\dfrac{dV}{dt}=kV\):
L3 Apply
Answer: Order 1 (only first derivative), degree 1 (linear in dV/dt).
Q2. Verify that \(V(t)=V_0 e^{kt}\) is the particular solution.
L3 Apply
Solution: \(dV/dt=kV_0 e^{kt}=kV\). ✓ Initial condition: \(V(0)=V_0\). ✓ Note general solution is \(V=Ce^{kt}\); particular fixes \(C=V_0\).
Q3. (T/F) "The general solution of a 3rd-order DE always contains exactly 3 arbitrary constants." Justify.
L5 Evaluate
True. By the existence-uniqueness theory: an n-th order DE's general solution carries n free parameters, fixed by n initial conditions. For 3rd-order, 3 constants.
Q4. Form the DE for the family \(y=Ax^2+Bx\) (two-parameter family).
L4 Analyse
Solution: Two parameters → 2nd order DE. \(y'=2Ax+B\); \(y''=2A\). From \(y''=2A\): \(A=y''/2\). From \(y'=2Ax+B\): \(B=y'-y''x\). Substitute into \(y=Ax^2+Bx\): \(y=(y''/2)x^2+(y'-y''x)x=(y'')x^2/2+y'x-y''x^2=y'x-(y''x^2)/2\). Rearrange: \(x^2 y''-2xy'+2y=0\).
Q5. Design: a tank loses water at a rate proportional to the height of water (Torricelli-like). Write the differential equation governing height h(t), and identify order/degree.
L6 Create
Solution: \(\dfrac{dh}{dt}=-k\,h\) (proportional, hence rate ∝ h; minus sign for decrease). Order 1, degree 1, linear. Solution: \(h(t)=h_0 e^{-kt}\) — exponential decay. (Real Torricelli is \(dh/dt\propto -\sqrt h\), nonlinear; this is a simpler analogue.)

Assertion–Reason Questions

Assertion (A): The DE \(\sin(y'')+y=x\) has degree undefined.
Reason (R): Degree is defined only for DEs that are polynomial in the derivatives.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Answer: (a). sin of a derivative is non-polynomial, so degree is undefined per R.
Assertion (A): A solution of a DE substituted into the DE produces an identity.
Reason (R): By definition of "solution", the equation must hold for all x in the relevant domain.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Answer: (a). Tautological — A is the definition; R is the precise statement.
Assertion (A): The DE \(\dfrac{d^2y}{dx^2}+y=0\) has \(y=\sin x\) as a particular solution.
Reason (R): The general solution is \(y=A\cos x+B\sin x\), and choosing A=0, B=1 gives \(y=\sin x\).
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Answer: (a). R is the construction that yields A.

Frequently Asked Questions — Differential Equations — Introduction, Order and Degree

What is a differential equation?
An equation involving derivatives of one or more dependent variables with respect to one or more independent variables.
What is the order of a differential equation?
The order is the highest derivative that appears.
What is the degree of a differential equation?
After clearing radicals/fractions, the degree is the highest power of the highest-order derivative — provided the equation is polynomial in derivatives. Otherwise undefined.
What is a general solution?
A solution containing as many arbitrary constants as the order of the differential equation.
What is a particular solution?
A solution obtained from the general solution by giving specific values to the arbitrary constants — typically via initial conditions.
How do you form a differential equation from a family of curves?
If the family has n arbitrary constants, differentiate n times and eliminate the constants to get an n-th order DE.
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