Master Exact Equations: Enhance Your Differential Equations Skills

Question 1

To solve an exact equation, the first step is to:

Accepted Answer

Check the exactness condition.

Answer

Find the potential function.

Answer

Separate variables.

Answer

Use an integrating factor.

Question 2

Which of the following is NOT a potential function for the exact equation ydx + (x + y^2)dy = 0?

Accepted Answer

None of the options is a potential function for this equation.

Answer

xy

Answer

x^2/2 + y^3/3

Answer

y^2/2 + xy

Question 3

The solution to the exact equation 2xy dx + (x^2 + y) dy = 0 is:

Accepted Answer

x^2y + xy^2 = C

Answer

y = Cx^2

Answer

x = Cy^2

Answer

xy = C

Question 4

A common application of exact equations is:

Accepted Answer

Modeling fluid flow

Answer

Predicting population growth

Answer

Calculating the area of a surface

Answer

Solving circuit problems

Question 5

The order of the exact equation x^2y' + y = xy is:

Accepted Answer

First order

Answer

Second order

Answer

Third order

Answer

Indeterminate

Question 6

A characteristic of homogeneous exact differential equations is that:

Accepted Answer

The coefficients of x and y are functions of x + y.

Answer

The coefficients of x and y are polynomials.

Answer

The coefficients of x and y are constant.

Answer

The equation is linear in x and y.

Question 7

Why does an exact differential equation imply the existence of a potential function?

Accepted Answer

An exact differential equation can be expressed as the total derivative of a potential function.

Answer

Exact differential equations guarantee solutions, while potential functions may not exist for all differential equations.

Answer

Potential functions are exclusive to first-order differential equations, unlike exact equations which can be of any order.

Question 8

Determine if the following equation is exact:

Accepted Answer

(2xy + y^2)dx + (x^2 + 2xy)dy = 0

Answer

(2xy - y^2)dx + (x^2 + 2xy)dy = 0

Answer

(2xy + y^2)dx - (x^2 + 2xy)dy = 0

Answer

(2xy - y^2)dx - (x^2 + 2xy)dy = 0

Question 9

Which of the following is NOT a necessary condition for an equation to be exact?

Accepted Answer

The equation is linear in both x and y.

Answer

The partial derivatives of M and N with respect to y and x are equal.

Answer

The coefficients of dx and dy are functions of x and y only.

Answer

The differential equation is of the form M(x, y)dx + N(x, y)dy = 0.

Question 10

Which of the following is NOT a step in the method of solving exact equations?

Accepted Answer

Find the integrating factor if the equation is not exact.

Answer

Check if the equation is exact.

Answer

Multiply the equation by the integrating factor.

Answer

Integrate the resulting equation.

Question 11

Find the integrating factor for the equation (x + 2y) dx + (x - y) dy = 0. An integrating factor is a function that makes an equation exact.

Accepted Answer

1/(x+2y)

Answer

1/x

Answer

2x

Answer

1/(x-y)

Question 12

Which of the following is the correct definition of an exact differential equation?

Accepted Answer

A differential equation that can be written in the form M(x,y)dx + N(x,y)dy = 0, where M and N have continuous partial derivatives in a region containing the point (x,y).

Answer

A differential equation that can be solved using the method of integrating factors.

Answer

A differential equation that has a solution that is a constant.

Answer

A differential equation that is linear in both x and y.

Question 13

Which of the following is the solution to an exact differential equation M(x,y)dx + N(x,y)dy = 0?

Accepted Answer

∫M(x,y)dx + ∫N(x,y)dy = C

Answer

∫(M(x,y) + N(x,y))dx + ∫(M(x,y) - N(x,y))dy = C

Answer

∫(M(x,y) - N(x,y))dx + ∫(M(x,y) + N(x,y))dy = C

Answer

∫(M(x,y)dx + N(x,y)dy) = C

Question 14

Consider the differential equation (2x + 3y)dx + (x + 3y)dy = 0. Is this equation exact?

Accepted Answer

Yes, because it can be written in the form M(x,y)dx + N(x,y)dy = 0 and satisfies the exactness condition.

Answer

No, because it cannot be written in the form M(x,y)dx + N(x,y)dy = 0.

Answer

No, because it does not satisfy the exactness condition.

Answer

Yes, because it is a first-order differential equation.

Question 15

Identify the exact equation from the following options:

Accepted Answer

y dy - x dx = 0

Answer

y' + x^2y = xy

Answer

dx^2 + dy^2 = 0

Answer

x^2 + y^2 = 1

Question 16

Which of the following conditions guarantees that a differential equation is exact?

Accepted Answer

Mixed partial derivatives are equal

Answer

Both partial derivatives of the function are continuous

Answer

The equation is linear

Answer

The equation is separable

Question 17

Which of the following techniques is used to solve non-exact equations?

Accepted Answer

Integrating factors

Answer

Method of undetermined coefficients

Answer

Separation of variables

Answer

Variation of parameters

Question 18

What is the integrating factor for the equation x dy - y dx = xe^x dx?

Accepted Answer

e^x

Answer

e^-x

Answer

x

Answer

1/x

Question 19

How do you determine if a non-exact equation can be made exact by multiplying by an appropriate integrating factor?

Accepted Answer

Check if the mixed partial derivatives are equal

Answer

Check if both partial derivatives exist and are continuous

Answer

Check if the equation is linear

Answer

Check if the equation is separable

Question 20

Which of the following is a limitation of the method for solving exact differential equations?

Accepted Answer

It requires the equation to be exact or reducible to exact.

Answer

It only applies to first-order equations.

Answer

It cannot handle non-linear equations.

Answer

It always provides a unique solution.

Question 21

Identify the exact differential equation:

Accepted Answer

M dx + N dy = 0, where M and N are functions of x and y

Answer

(x + y) dx + (x - y) dy = 0

Answer

x dy - y dx = 0

Answer

M(x, y) dx + N(x, y) dy = 0, where M and N are constants

Question 22

Which of the following differential equations is **exact**?

Accepted Answer

(x + y^2)dx + (2xy - e^y)dy = 0

Answer

(x^2 + y^2)dx + (2xy)dy = 0 (Not exact)

Answer

(xy + y)dx + (x^2 + 2y)dy = 0 (Not exact)

Answer

xdx + ydy = 1 (Not exact)

Question 23

Find an **integrating factor** for the equation (2x + 3y)dx + (x - y)dy = 0.

Accepted Answer

μ(x, y) = xy

Answer

μ(x, y) = 1/(x^2 + y^2) (Incorrect)

Answer

μ(x, y) = e^(x+y) (Incorrect)

Answer

μ(x, y) = x^2 + y^2 (Incorrect)

Question 24

Solve the **exact** equation (x^2 + y^2)dx + 2xydy = 0.

Accepted Answer

x^2 + y^2 = C

Answer

x - y = C (Incorrect)

Answer

x^2 - y^2 = C (Incorrect)

Question 25

Determine whether the equation (y + sin(x))dx + (x + cos(y))dy = 0 is **exact**.

Accepted Answer

Not exact

Answer

Exact (Incorrect)

Answer

Exact but only for x > 0 (Incorrect)

Answer

Cannot be determined (Incorrect)

Question 26

Find the general solution of the **exact** equation ye^(x+y)dx + (xe^(x+y) + 2y)dy = 0.

Accepted Answer

xy e^(x+y) = C

Answer

x^2 e^(x+y) + y^2 = C (Incorrect)

Answer

e^(x+y) = C (Incorrect)

Question 27

Which of the following is a **necessary** condition for a differential equation Mdx + Ndy = 0 to be exact?

Accepted Answer

∂M/∂y = ∂N/∂x

Answer

M = N (Incorrect)

Answer

∂M/∂x + ∂N/∂y = 0 (Incorrect)

Question 28

Consider the equation (2x + y)dx + (x - 3y)dy = 0. Find an **implicit solution**.

Accepted Answer

x^2 + xy - 3y^2 = C

Answer

xy - 3y^2 = C (Incorrect)

Answer

x - y = C (Incorrect)

Answer

x^2 + y^2 = C (Incorrect)

Question 29

Which of the following differential equations is **NOT** exact?

Accepted Answer

xy^2 dx + (x^2y + sin(x))dy = 0

Answer

(x^2 + y^2)dx + 2xydy = 0 (Exact)

Answer

e^y dx + (xe^y - y)dy = 0 (Exact)

Answer

(y - x^2)dx + (x - y^2)dy = 0 (Exact)

Question 30

Solve the **exact** equation (x^3 + y^3)dx + (3x^2y - xy^2)dy = 0.

Accepted Answer

x^3y - xy^3 = C

Answer

x^2 + y^2 = C (Incorrect)

Answer

xy = C (Incorrect)

Answer

x^3 - y^3 = C (Incorrect)

Question 31

Solve the exact equation (y + 2xy^2)dx + (2x^2y + 2y)dy = 0.

Accepted Answer

x^2y + y = C

Answer

2x^2y + y = C

Answer

x^2y - y = C

Question 32

Consider the differential equation xydx + (y - x^2)dy = 0. Determine if it is exact and find its general solution.

Accepted Answer

x^2y - y^2 = C

Answer

xy^2 - x^2y = C

Answer

y - x^2 = C

Answer

2x^2y - y^2 = C

Question 33

A differential equation is considered exact if it satisfies a specific condition. Which of the following conditions defines an exact differential equation?

Accepted Answer

∂M/∂y = ∂N/∂x, where M(x, y)dx + N(x, y)dy = 0

Answer

∂M/∂y = ∂N/∂x + 1, where M(x, y)dx + N(x, y)dy = 0

Answer

∂M/∂x = ∂N/∂x, where M(x, y)dx + N(x, y)dy = 0

Answer

∂M/∂x = ∂N/∂y, where M(x, y)dx + N(x, y)dy = 0

Question 34

The general solution to the exact differential equation (2xy + 1)dx + (x² + 3y²)dy = 0 can be expressed in the form F(x, y) = C. What is the correct form of F(x, y)?

Accepted Answer

x²y + x + y³

Answer

x²y² + x + y³

Answer

x²y + x + 2y³

Answer

x²y + 2x + y³

Question 35

When solving an exact differential equation, which of the following steps is NOT typically required?

Accepted Answer

Applying the method of undetermined coefficients.

Answer

Verifying if the equation satisfies the exactness condition.

Answer

Integrating to find the general solution.

Answer

Finding an integrating factor if necessary.

Question 36

What is the general solution to the exact differential equation 2xydx + (x² + y²)dy = 0?

Accepted Answer

x²y + y²/2 = C

Answer

x²y² + y²/2 = C

Answer

x²y + y³ = C

Answer

x²y + y = C

Question 37

Consider the differential equation M(x, y)dx + N(x, y)dy = 0. If ∂M/∂y = ∂N/∂x, what can we conclude about this differential equation?

Accepted Answer

The differential equation is exact.

Answer

The differential equation is homogeneous.

Answer

The differential equation is separable.

Answer

The differential equation is linear.

Question 38

Suppose we find a potential function Φ(x, y) for an exact differential equation, where ∂Φ/∂x = M and ∂Φ/∂y = N. What represents the general solution to this differential equation?

Accepted Answer

Φ(x, y) = C

Answer

M + N = 0

Answer

Φ(x, y) = 0

Answer

∂Φ/∂x + ∂Φ/∂y = 0

Question 39

Which of the following is NOT a benefit of using exact differential equations to model phenomena?

Accepted Answer

They are always easier to solve than other types of differential equations.

Answer

They can provide a more accurate representation of certain physical systems.

Answer

They can be used to model various physical and engineering systems.

Answer

They offer a systematic method for finding solutions.

Question 40

If a differential equation is not exact, is there a way to potentially make it exact?

Accepted Answer

Yes, it is sometimes possible to find an integrating factor to make it exact.

Answer

No, it's impossible to make a non-exact equation exact.

Answer

Yes, but only if the equation is homogeneous.

Question 41

A differential equation of the form M(x, y) dx + N(x, y) dy = 0 is considered exact if which of the following conditions holds true?

Accepted Answer

∂M/∂y = ∂N/∂x

Answer

∂M/∂x = ∂N/∂y

Answer

∂M/∂y = -∂N/∂x

Answer

∂M/∂x = -∂N/∂y

Question 42

What is the integrating factor for the differential equation (2xy + y²) dx + (x² + 2xy) dy = 0?

Accepted Answer

1

Answer

y

Answer

xy

Answer

x

Question 43

What is the general solution of the exact differential equation (2xy + y²) dx + (x² + 2xy) dy = 0?

Accepted Answer

x²y + xy² = C

Answer

x²y + y² = C

Answer

x²y + xy³ = C

Answer

x³y + xy² = C

Question 44

If the partial derivative of M with respect to y is not equal to the partial derivative of N with respect to x (∂M/∂y ≠ ∂N/∂x), what can we conclude about the differential equation M(x, y) dx + N(x, y) dy = 0?

Accepted Answer

The equation is not exact.

Answer

The equation is linear.

Answer

The equation is exact.

Answer

The equation is separable.

Question 45

Is the differential equation (y² + 2xy) dx + (x² + 2xy) dy = 0 an exact differential equation?

Accepted Answer

Yes, it is exact.

Answer

It is linear.

Answer

It is separable.

Answer

No, it is not exact.

Question 46

If the integrating factor for a differential equation is μ(x, y) = x, what happens to the equation after multiplying by this integrating factor?

Accepted Answer

The equation becomes exact.

Answer

The equation remains unchanged (it is already exact).

Answer

The equation becomes separable.

Answer

The equation becomes linear.

Question 47

What type of differential equation is represented by y' + p(x)y = q(x)?

Accepted Answer

A linear first-order differential equation.

Answer

A homogeneous differential equation.

Answer

A separable differential equation.

Answer

An exact differential equation.

Question 48

What is the integrating factor for the differential equation (x² + y²) dx - 2xy dy = 0?

Accepted Answer

1/x²

Answer

y²

Answer

x²

Answer

1/y²

Question 49

After multiplying the differential equation (x² + y²) dx - 2xy dy = 0 by its integrating factor, what type of equation does it become?

Accepted Answer

An exact differential equation.

Answer

A separable differential equation.

Answer

A non-exact differential equation.

Answer

A linear differential equation.

Question 50

Which of the following is a necessary condition for a differential equation of the form P(x, y) dx + Q(x, y) dy = 0 to be exact?

Accepted Answer

The partial derivative of P with respect to y is equal to the partial derivative of Q with respect to x (∂P/∂y = ∂Q/∂x).

Answer

The general solution is constant.

Question 51

A differential equation is exact if it can be written in the form:?

Accepted Answer

M(x,y)dx + N(x,y)dy = 0

Answer

M(x,y)dy + N(x,y)dx = 0

Answer

M(y)dx + N(x)dy = 0

Answer

M(x)dx + N(y)dy = 0

Question 52

What is the integrating factor for the differential equation: (2x - 3y)dx + (2y - 3x)dy = 0?

Accepted Answer

1/(x - y)

Answer

1/(x + y)

Answer

x + y

Answer

x - y

Question 53

Which of the following is a limitation of the exact equations method?

Accepted Answer

It only applies to first-order equations

Answer

It is not applicable to equations with constant coefficients

Answer

It can lead to complex integrations

Answer

It requires the equation to be homogeneous

Question 54

Consider the differential equation: y' + P(x)y = Q(x). Which of the following is true about the integrating factor?

Accepted Answer

It depends on P(x) only

Answer

It is always 1

Answer

It depends on both P(x) and Q(x)

Answer

It depends on y only

Question 55

What is the integrating factor for the differential equation: (y + x^2y^3)dx + (x + x^2y^3)dy = 0?

Accepted Answer

e^(x^2y^2)

Answer

1/(x^2y^2)

Answer

x^2y^2

Answer

e^(-x^2y^2)