Differential Equations

QUESTION: 1

Let a, b ∈ R. Let y = (y₁, y₂)^T be a solution of the system of equations

Every solution y(x) → 0 as x → ∞ if

A.
a < 0, b < 0
B.
a < 0, b > 0
C.
a > 0, b > 0
D.
a > 0 , b < 0

Solution:

QUESTION: 2

The solution of exists for all

A.
x ∈ (-∝, 1)
B.
x ∈ [0, a] where a > 1
C.
x ∈ (-∝, ∝)
D.
x ∈ [1, a] where a > 1

Solution:

QUESTION: 3

For solving by variation of parameters. The value of Wronskion W is

A.
1
B.
2
C.
3
D.
4

Solution:

QUESTION: 4

The initial value problem

A.
a unique solution
B.
no solution
C.
infinitely many solutions
D.
two linearly independent solution

Solution:

QUESTION: 5

Consider the following statements
I. A singular solution of differential equation satisfies the differential equation but is not a particular solution of the equation.
II. If T(x, y) = 0 is the equation of the tac-locus, then
T(x, y ) is a factor of the P-discriminant.
III. T(x, y ) is a factor of a c-discriminant.
IV. Cusp-locus has two distinct tangent.
Choose the correct answer.

A.
Only I is true
B.
I, II and IV are true
C.
All are true
D.
No one is true

Solution:

QUESTION: 6

The general solution
where andy(0) = 0 is

A.
B.
C.
D.
None of the above

Solution:

QUESTION: 7

Which of the following pair of function are linear independent

Select the correct code

A.
Only I is true
B.
Only II is true
C.
I and III are true
D.
All are true

Solution:

QUESTION: 8

Singular solution of differential equation contains
I. arbitrary constant
II. can be obtained from general
III. do not contain arbitrary constant
IV. cannot be obtained from general solution

A.
I, II are true
B.
III, IV are true
C.
I, IV are true
D.
II, IV are true

Solution:

QUESTION: 9

For non-homogeneous equation y' + p(x)y = r(x), if y₁ and y₂ are its solutions, then the solution of homogeneous equation y' + p(x)y = 0 is

A.
y = y_{1 -} y₂
B.
C.
D.
None of these

Solution:

QUESTION: 10

The solution of represents

A.
a family of circle centre at (1, 0)
B.
a family of circle centre at (0, 0)
C.
a family of circle centre at ( - 1 , 0)
D.
a family of straight line with slope -1

Solution:

QUESTION: 11

Consider the differential equation (x + y + 1) dx + (2x + 2y + 1)dy = 0. Which of the following statement is true?

A.
A suitable substitution transform the differentiable equation to the variables separable from
B.
The differential equation is linear
C.
The differential equation is exact
D.
ex + y is an integrating factor of the differential equation

Solution:

Here (x + y + 1)dx + (2x + 2y + 1)dy = 0

By separables of variables,

Integrating (2z + log z) = x + c
⇒ 2(x + y) + log (x + y) = x + c.

QUESTION: 12

The orthogonal trajectory of the family x² - y² = C₂ are given by

A.
x² + y² = C₂
B.
xy = C₂
C.
y = C₂
D.
x = 0

Solution:

QUESTION: 13

A. Singular solution contains no arbitrary constants.
B. Singular solution can be obtained from complete primitive.

A.
A is true, B is false
B.
Both A and B are true
C.
Both A and B are false
D.
None of the above

Solution:

QUESTION: 14

Let the general solution of a differential equation be, y = ae^bx+c then order of the differential equation is,

A.
1
B.
2
C.
3
D.
can not say

Solution:

QUESTION: 15

If y₁(x) = x and y₂(x) = xe^x are two linearly independent solutions of then the interval on which they form a fundam ental set of solution is

A.
x > 0 or x < 0
B.
-1 < x < ∝
C.
- 1 < x < 2
D.
-∝ < x < ∝

Solution:

QUESTION: 16

Using the method of variation of parameters for the particular solution to the differential equation

Solution:

QUESTION: 17

The general solution of the system of differential equation and M a 2 x 2 matrix and a 2 x 1 constant vector b = is given by

A.
e^MT c + b
B.
e^mt c + bt
C.
e^MT c - b
D.
e^MT c + bt

Solution:

QUESTION: 18

For the ordinary differential equation

which of the following statement is true?

A.
0 is regular and 1 is irregular.
B.
0 is irregular and 1 is regular.
C.
Both 0 and 1 are regular.
D.
Both 0 and 1 are irregular,

Solution:

QUESTION: 19

If 2x(1 - y ) = K and g(x, y) = L are orthogonal families of curves where K and L constants, then g(x, y) is

A.
x² + 2y - y²
B.
2{y + 2(y + (1 - x))}
C.
x² + 2x - y²
D.
x² - 2y - y²

Solution:

QUESTION: 20

If y₁and y₂ are linearly independent solutions of the homogeneous equation L(y) =y"+p₁(x)y' + p₂(x)y = 0. Then, p₁(x) and p₂{x) are given by

Solution:

Differential Equations - 4

20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Differential Equations - 4

Let a, b ∈ R. Let y = (y₁, y₂)^T be a solution of the system of equations

Every solution y(x) → 0 as x → ∞ if

The solution of exists for all

For solving by variation of parameters. The value of Wronskion W is

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