The order of d²y/dx² + 3 dy/dx + y = 0 is:

2. The highest derivative is the second.

The degree of (d²y/dx²)³ + (dy/dx)⁴ = x is:

3. The power of the highest order derivative d²y/dx² is 3.

The order and degree of dy/dx = x + y are:

1, 1. First derivative to the first power.

The number of arbitrary constants in the general solution equals the:

Order. Each integration introduces a constant.

Maharashtra State Board · HSC Commerce · Class 12 (Std XII)

Class 12 Maths and Statistics Part 1 Chapter 8: Differential Equation and Applications

Short notes, key concepts, quick revision and practice MCQs with answers

A differential equation involves derivatives of an unknown function. The chapter defines order and degree, forms differential equations by eliminating arbitrary constants, solves variable separable, homogeneous and linear equations, and applies them to growth and decay problems.

Key concepts: Differential Equation and Applications

Differential equation: An equation involving the derivatives of a dependent variable with respect to an independent variable. If there is one independent variable, it is an ordinary differential equation.
Order and degree: Order: the order of the highest derivative present. Degree: the power of the highest order derivative after the equation is made free of radicals and fractional powers of derivatives. Both are positive integers.
Solution: A general solution contains as many arbitrary constants as the order. A particular solution is obtained by giving particular values to the constants, usually from given conditions.
Formation: To form the differential equation of a family of curves with n arbitrary constants, differentiate n times and eliminate the constants. The order equals the number of constants.
Variable separable: If the equation can be written as f(x) dx = g(y) dy, integrate both sides: ∫f(x) dx = ∫g(y) dy + c.
Homogeneous equations: f(x, y) dx + g(x, y) dy = 0 where f and g are homogeneous of the same degree. Put y = vx, dy/dx = v + x dv/dx, and separate the variables.
Linear equations: dy/dx + Py = Q (P, Q functions of x). Integrating factor I.F. = e^(∫P dx); solution y(I.F.) = ∫Q(I.F.) dx + c. If linear in x: dx/dy + Px = Q, solution x(I.F.) = ∫Q(I.F.) dy + c with I.F. = e^(∫P dy).
Applications: Growth and decay: if the rate of change of a quantity is proportional to the quantity, dx/dt = kx, so x = x₀ e^(kt) (k > 0 growth, k < 0 decay). Used for population, bacteria, and value of assets.

Quick revision: important points

Degree is defined only when the equation is a polynomial in its derivatives.
dy/dx = y gives y = c eˣ; dy/dx = ky gives y = c e^(kx).
For dy/dx + y/x = Q, I.F. = e^(log x) = x.
The number of arbitrary constants in the general solution equals the order.
Particular solutions use given initial conditions.

Practice MCQs with answers

The order of d²y/dx² + 3 dy/dx + y = 0 is:
1. 1
2. 2
3. 3
4. 0
Show answer
Answer: B. 2
The highest derivative is the second.
The degree of (d²y/dx²)³ + (dy/dx)⁴ = x is:
1. 2
2. 3
3. 4
4. 1
Show answer
Answer: B. 3
The power of the highest order derivative d²y/dx² is 3.
The order and degree of dy/dx = x + y are:
1. 1, 1
2. 1, 2
3. 2, 1
4. 0, 1
Show answer
Answer: A. 1, 1
First derivative to the first power.
The number of arbitrary constants in the general solution equals the:
1. Degree
2. Order
3. Order + degree
4. One always
Show answer
Answer: B. Order
Each integration introduces a constant.
The general solution of dy/dx = 2x is:
1. y = 2 + c
2. y = x² + c
3. y = 2x² + c
4. y = x + c
Show answer
Answer: B. y = x² + c
Integrate 2x.

All Class 12 Maths and Statistics chapters

Class 12 Maths and Statistics notes: all chapters