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# Prove a method to obtain a first-order, separable differential equation

Prove that the substitution transforms a homogeneous equation into a first-order, separable equation for .

Proof. Let Since is homogeneous, we have Letting we then have So, Hence, this equation is separable # Find an implicit formula satisfied by solutions of y y′ = ex + 2y sin x

Assume solutions of the equation exist and find an implicit formula satisfied by these solutions.

This is a first order separable equation. We compute Both of the integrals were evaluated using integration by parts. Also, we already established a formula for in this exercise (Section 6.17, Exercise #20, taking ).

# Find an implicit formula satisfied by solutions of xyy′ = 1 + x2 + y2 + x2y2

Assume solutions of the equation exist and find an implicit formula satisfied by these solutions.

This is a first-order separable equation. We compute # Find an implicit formula satisfied by solutions of (x2 – 4) y′ = y

Assume solutions of the equation exist and find an implicit formula satisfied by these solutions.

This a first-order separable equation. We compute # Find an implicit formula satisfied by solutions of xy(1 + x2) y′ – (1 + y2) = 0

Assume solutions of the equation exist and find an implicit formula satisfied by these solutions.

This is a separable first-order equation. We compute # Find an implicit formula satisfied by solutions of (1 – x2)1/2 y′ + 1 + y2 = 0

Assume solutions of the equation exist and find an implicit formula satisfied by these solutions.

This is a separable first-order equation. We compute # Find an implicit formula satisfied by solutions of (x – 1) y′ = xy

Assume solutions of the equation exist and find an implicit formula satisfied by these solutions.

This is a separable first-order equation. We compute # Find an implicit formula satisfied by solutions of y (1 – x2)1/2 y′ = x

Assume solutions of the equation exist and find an implicit formula satisfied by these solutions.

This is a separable first order equation. We compute # Find an implicit formula satisfied by solutions of y′ = (y-1)(y-2)

Assume solutions of the equation exist and find an implicit formula satisfied by these solutions.

This is a separable first order equation. We compute # Find an implicit formula satisfied by solutions of (x + 1) y′ + y2 = 0

Assume solutions of the equation exist and find an implicit formula satisfied by these solutions.

This is a separable first order equation. We compute 