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

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Tag: Ordinary Differential Equations

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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

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 ).

Assume solutions of the equation

exist and find an implicit formula satisfied by these solutions.

This is a first-order separable equation. We compute

Assume solutions of the equation

exist and find an implicit formula satisfied by these solutions.

This a first-order separable equation. We compute

Assume solutions of the equation

exist and find an implicit formula satisfied by these solutions.

This is a separable first-order equation. We compute

Assume solutions of the equation

exist and find an implicit formula satisfied by these solutions.

This is a separable first-order equation. We compute

Assume solutions of the equation

exist and find an implicit formula satisfied by these solutions.

This is a separable first-order equation. We compute

Assume solutions of the equation

exist and find an implicit formula satisfied by these solutions.

This is a separable first order equation. We compute

Assume solutions of the equation

exist and find an implicit formula satisfied by these solutions.

This is a separable first order equation. We compute

Assume solutions of the equation

exist and find an implicit formula satisfied by these solutions.

This is a separable first order equation. We compute