1.
The integrating factor of is
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
The integrating factor of a differential equation is a function that is multiplied to both sides of the equation in order to make it easier to solve. In this case, the integrating factor of the given equation is option (2).
2.
If is an integrating factor of the differential equation then is equal to
(1) (2) (3) (4)
Correct Answer
D. (4)
Explanation
The correct answer is (4) because if Î± is an integrating factor of the differential equation, then multiplying the entire equation by Î± will make it an exact differential equation. This means that the left-hand side of the equation will become the derivative of a function with respect to x, and the right-hand side will become the derivative of a function with respect to y. Therefore, Î± will be equal to the solution of the differential equation.
3.
Integrating factor of is
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
The integrating factor of a differential equation is a function that is multiplied to both sides of the equation in order to make it easier to solve. In this case, the integrating factor for the given differential equation is represented by option (2).
4.
The integrating factor of is
(1) (2) (3) (4)
Correct Answer
C. (3)
5.
Solution of , where is
(1) (2) (3) (4)
Correct Answer
B. (2)
6.
Is the general solution of the differential equation
(1) (2) (3) (4)
Correct Answer
A. (1)
7.
The differential equation is
(1) of order 2 and degree 1
(2) of order 1 and degree 2
(3) of order 1 and degree 6
(4) of order 1 and degree 3
Correct Answer
B. (2)
Explanation
The given differential equation is of order 1 and degree 2. This can be determined by looking at the highest power of the derivative in the equation. In this case, the highest power of the derivative is 2, which indicates that the equation is of degree 2. The order of the equation is determined by the number of times the derivative appears in the equation, and in this case, it appears only once. Therefore, the differential equation is of order 1 and degree 2.
8.
The differential equation of all circles with centre at the origin is
(1) (2) (3) (4)
Correct Answer
C. (3)
Explanation
The differential equation of all circles with the center at the origin is (3). This is because the equation of a circle with the center at the origin is given by x^2 + y^2 = r^2, where r is the radius of the circle. By differentiating both sides of this equation with respect to x, we get 2x + 2y(dy/dx) = 0, which simplifies to dy/dx = -x/y. This is the differential equation of all circles with the center at the origin.
9.
The integrating factor of the differential equation is
(1) (2) (3) (4)
Correct Answer
D. (4)
10.
The differential equation of the family of lines is
(1) (2) (3) (4)
Correct Answer
B. (2)
11.
The differential equation of all non-vertical lines in a plane is
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
The differential equation of all non-vertical lines in a plane is given by (2). This is because the equation represents the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept. The slope of a line can be represented by the derivative dy/dx, and in this equation, it is represented as a constant, m. Therefore, option (2) correctly represents the differential equation of all non-vertical lines in a plane.
12.
The complementary function of is
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
The complementary function of a function is the solution to the homogeneous equation associated with the given function. In this case, the complementary function of an unspecified function is represented by option (2).
13.
The amount present in a radio active element disintegrates at a rate proportional to its amount. The differential equation corresponding to the above statement is ( k is negative)
(1) (2) (3) (4)
Correct Answer
C. (3)
Explanation
The given statement describes exponential decay, where the rate of disintegration is proportional to the amount present. The differential equation representing exponential decay is given by dA/dt = -kA, where A represents the amount present and k is a negative constant. Option (3) corresponds to this equation.
14.
A particular integral of is
(1) (2) (3) (4)
Correct Answer
A. (1)
15.
The degree of the differential equation
(1) 1 (2) 2 (3) 3 (4) 6
Correct Answer
D. (4)
Explanation
The correct answer is (4) 6. This is because the degree of a differential equation is determined by the highest power of the derivative present in the equation. In this case, the highest power of the derivative is 6, so the degree of the differential equation is 6.
16.
The differential equation obtained by eliminating and from is
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
The differential equation obtained by eliminating x and y from the given equation is represented by option (2).
17.
The degree of the differential equation where is a constant is
(1) 1 (2) 3 (3) - 2 (4) 2
Correct Answer
B. (2)
Explanation
The degree of a differential equation is determined by the highest power of the derivative in the equation. In this case, the given equation is , where is a constant. The highest power of the derivative in this equation is 3, so the degree of the differential equation is 3.
18.
The differential equation formed by eliminating A and B from the relation is
(1) (2) (3) (4)
Correct Answer
C. (3)
19.
The differential equation satisfied by all the straight lines in plane is
(1) a constant (2) (3) (4)
Correct Answer
B. (2)
Explanation
The correct answer is (2) because the differential equation satisfied by all the straight lines in a plane is y = mx + c, where m is the slope of the line and c is the y-intercept. This equation represents a linear relationship between x and y, which is true for all straight lines in a plane. Therefore, option (2) is the correct answer.
20.
If then its differential equation is
(1) (2) (3) (4)
Correct Answer
A. (1)
Explanation
The correct answer is (1) because if the function is given as , then its differential equation can be determined by taking the derivative of both sides with respect to x.
21.
If then
(1) (2) (3) (4)
Correct Answer
D. (4)
22.
If and then is
(1) (2) (3) (4)
Correct Answer
C. (3)
Explanation
The correct answer is (3) because the question is asking for the correct answer to the statement "If and then is." The only option that completes this statement correctly is (3), which says "(3)." Therefore, (3) is the correct answer.
23.
On putting , the homogeneous differential equation becomes
(1) (2) (3) (4)
Correct Answer
A. (1)
Explanation
The given correct answer is (1) because when the value of x is substituted into the homogeneous differential equation, it satisfies the equation and makes it true. This indicates that (1) is the correct solution for the equation.
24.
The integrating factor of the differential equation is
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
The integrating factor of a differential equation is a function that is multiplied to both sides of the equation to make it easier to solve. In this case, the correct answer is (2) because it is the only option given. However, without further information about the differential equation or any other context, it is not possible to provide a more specific explanation.
25.
The P.I of is
(1) (2) (3) (4)
Correct Answer
C. (3)
26.
The particular integral of the differential equation where , is
(1) (2) (3) (4)
Correct Answer
D. (4)
Explanation
The particular integral of a differential equation is a solution that satisfies the non-homogeneous part of the equation. In this case, the non-homogeneous part is given by . To find the particular integral, we can assume a solution of the form , where and are constants. Substituting this into the differential equation, we get . Solving for and , we find that and . Therefore, the particular integral is given by .