12 - Maths - Unit 8 - Differential Equations

1. The complementary function of

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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).

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).

2. The integrating factor of the differential equation

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

3. Solution of

, where

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

4. The differential equation of all non-vertical lines in a plane is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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.

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.

5. Integrating factor of

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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).

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).

6. The particular integral of the differential equation

where

,

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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 .

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 .

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

(1)

(2)

(3)

(4)

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.

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. If

and

then

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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.

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.

9. The differential equation satisfied by all the straight lines in

plane is (1)

a constant (2)

(3)

(4)

(1)

(2)

(3)

(4)

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.

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.

10. 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)

(1)

(2)

(3)

(4)

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.

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.

11. The differential equation obtained by eliminating

and

from

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The differential equation obtained by eliminating x and y from the given equation is represented by option (2).

Explanation

The differential equation obtained by eliminating x and y from the given equation is represented by option (2).

12. If

then its differential equation is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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.

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.

13. A particular integral of

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

14. The degree of the differential equation

where

is a constant is (1) 1 (2) 3 (3) - 2 (4) 2

(1)

(2)

(3)

(4)

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.

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.

15. The integrating factor of the differential equation

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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.

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.

16.

is the general solution of the differential equation (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

17. The differential equation formed by eliminating A and B from the relation

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

18. The P.I of

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

19. If

is an integrating factor of the differential equation

then

is equal to (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The correct answer is (4) because if α is an integrating factor of the differential equation, then multiplying the entire equation by α 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, α will be equal to the solution of the differential equation.

Explanation

The correct answer is (4) because if α is an integrating factor of the differential equation, then multiplying the entire equation by α 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, α will be equal to the solution of the differential equation.

20. The degree of the differential equation

(1) 1 (2) 2 (3) 3 (4) 6

(1)

(2)

(3)

(4)

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.

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.

21. On putting

, the homogeneous differential equation

becomes (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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.

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.

22. The differential equation of all circles with centre at the origin is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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.

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.

23. If

then (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

not-available-via-ai

Explanation

not-available-via-ai

24. The integrating factor of

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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).

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).