Differential Equation Essentials Quiz

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Exact vs non-exact DE concepts

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Ihteram Ali

Pakistan

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Froggy Jumps Differential Equation Essentials QuizOnline version Exact vs non-exact DE concepts by Ihteram Ali 1 What is an exact differential equation in the form M(x,y)dx + N(x,y)dy = 0? a ∂M/∂y = ∂N/∂x b M depends only on x c M and N are constants 2 How do you make a non-exact differential equation exact? a Set N=0 b Multiply by an integrating factor μ(x,y) to satisfy ∂(μM)/∂y = ∂(μN)/∂x c Divide by M 3 What indicates an exactness mismatch in Mdx + Ndy = 0? a ∂M/∂y ≠ ∂N/∂x b M=N everywhere c Both derivatives are zero 4 What is an integrating factor in the context of exact DEs? a A function μ that makes μM dx + μN dy exact b A solution function c A constant multiplier 5 What is the quick test for exactness? a Check if M and N are separable b Check if M=N c Check if ∂M/∂y equals ∂N/∂x 6 Is this DE exact? M=2xy, N=x^2. Are ∂M/∂y and ∂N/∂x equal? a Yes, both are x b No, they are 0 and 2x c Yes, both are 2x 7 Once an DE is exact, what is the next step to solve it? a Find a potential function ψ with ψ_x = M and ψ_y = N b Integrate N with respect to y only c Solve for y(x) explicitly 8 What is the potential function in an exact DE? a A constant of integration only b A function ψ(x,y) with dψ = M dx + N dy c A function independent of x and y 9 Why are exact equations useful in solving DEs? a They require no initial conditions b They always have closed-form x(t) c They integrate to a single constant via ψ(x,y)=C 10 When does an integrating factor μ(x) depend only on x? a When N is zero b If (∂M/∂y − ∂N/∂x)/N is a function of x only c When M is zero

Froggy Jumps

Differential Equation Essentials QuizOnline version

Exact vs non-exact DE concepts

by Ihteram Ali

What is an exact differential equation in the form M(x,y)dx + N(x,y)dy = 0?

∂M/∂y = ∂N/∂x

M depends only on x

M and N are constants

How do you make a non-exact differential equation exact?

Set N=0

Multiply by an integrating factor μ(x,y) to satisfy ∂(μM)/∂y = ∂(μN)/∂x

Divide by M

What indicates an exactness mismatch in Mdx + Ndy = 0?

∂M/∂y ≠ ∂N/∂x

M=N everywhere

Both derivatives are zero

What is an integrating factor in the context of exact DEs?

A function μ that makes μM dx + μN dy exact

A solution function

A constant multiplier

What is the quick test for exactness?

Check if M and N are separable

Check if M=N

Check if ∂M/∂y equals ∂N/∂x

Is this DE exact? M=2xy, N=x^2. Are ∂M/∂y and ∂N/∂x equal?

Yes, both are x

No, they are 0 and 2x

Yes, both are 2x

Once an DE is exact, what is the next step to solve it?

Find a potential function ψ with ψ_x = M and ψ_y = N

Integrate N with respect to y only

Solve for y(x) explicitly

What is the potential function in an exact DE?

A constant of integration only

A function ψ(x,y) with dψ = M dx + N dy

A function independent of x and y

Why are exact equations useful in solving DEs?

They require no initial conditions

They always have closed-form x(t)

They integrate to a single constant via ψ(x,y)=C

When does an integrating factor μ(x) depend only on x?

When N is zero

If (∂M/∂y − ∂N/∂x)/N is a function of x only

When M is zero

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Differential Equation Essentials Quiz

Froggy Jumps

About this activity

Created by

Download the paper version to play

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Differential Equation Essentials QuizOnline version

by Ihteram Ali

What is an exact differential equation in the form M(x,y)dx + N(x,y)dy = 0?

How do you make a non-exact differential equation exact?

What indicates an exactness mismatch in Mdx + Ndy = 0?

What is an integrating factor in the context of exact DEs?

What is the quick test for exactness?

Is this DE exact? M=2xy, N=x^2. Are ∂M/∂y and ∂N/∂x equal?

Once an DE is exact, what is the next step to solve it?

What is the potential function in an exact DE?

Why are exact equations useful in solving DEs?

When does an integrating factor μ(x) depend only on x?

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