Vidaara.orgClass 12 · Mathematics
CodeVID-M12-09-ODF-01
Order, Degree & Formation — Assignment
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
The degree of $\left(\dfrac{d^2y}{dx^2}\right)^3+\dfrac{dy}{dx}=0$ is:
- A.$1$
- B.$2$
- C.$3$
- D.undefined
2.
The order of $\dfrac{dy}{dx}+y=x$ is:
- A.$1$
- B.$2$
- C.$0$
- D.$3$
3.
The differential equation of the family $y=mx$ is:
- A.$y=x$
- B.$x\dfrac{dy}{dx}=y$
- C.$\dfrac{dy}{dx}=x$
- D.$y\dfrac{dy}{dx}=x$
4.
A general solution of an order-$n$ DE has:
- A.$1$ constant
- B.$n$ constants
- C.$n-1$ constants
- D.no constant
5.
Degree is defined only when the equation is:
- A.linear
- B.polynomial in its derivatives
- C.first order
- D.homogeneous
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find the order and degree of $\dfrac{d^2y}{dx^2}+3\left(\dfrac{dy}{dx}\right)^2+y=0$.
7.
Find the order and degree of $\left(\dfrac{dy}{dx}\right)^3+2y=x$.
8.
Verify that $y=e^{2x}$ is a solution of $\dfrac{dy}{dx}=2y$.
9.
Form the differential equation of the family $y=cx$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find the order and degree of $\left(\dfrac{d^2y}{dx^2}\right)^2+\left(\dfrac{dy}{dx}\right)^3+y=0$.
11.
Form the differential equation of the family $y=mx+c$.
12.
Form the differential equation of $y=Ae^{x}+Be^{-x}$.
13.
Verify that $y=A\cos x+B\sin x$ satisfies $y''+y=0$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Form the differential equation of the family of circles $x^2+y^2=a^2$.
15.
Form the differential equation representing $y=ae^{3x}+be^{-2x}$.
Answer Key
Section A — Multiple Choice Questions
- (C) $3$
- (A) $1$
- (B) $x\dfrac{dy}{dx}=y$
- (B) $n$ constants
- (B) polynomial in its derivatives
Section B — Short Answer (2 marks)
- Order $2$, degree $1$.
- Order $1$, degree $3$.
- Yes, it is a solution.
- $x\dfrac{dy}{dx}=y$.
Section C — Short Answer (3 marks)
- Order $2$, degree $2$.
- $\dfrac{d^2y}{dx^2}=0$.
- $\dfrac{d^2y}{dx^2}-y=0$.
- Yes, it satisfies it.
Section D — Long Answer (5 marks)
- $x+y\dfrac{dy}{dx}=0$.
- $\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}-6y=0$.
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