Ch 04/Variable Separable Method

Variable Separable Method

Rearrange so all y terms go left, all x terms go right, then integrate both sides.

IThe Separable Method

A DE is variable separable if it can be written as:

Once separated, integrate both sides independently:

When can we separate? When can be written as a product — then divide both sides by and multiply by .

Note

The constant C only needs to appear on one side. Whether you write , , or depends on which form gives the cleanest answer.

Example 1easy

Solve .

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Rewrite to see the product form

— separable since it's

Separate variables

Multiply both sides by , multiply by :

Integrate left side

Integrate right side

Combine (absorb 2 into C)

Example 2easy

Solve .

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Separate

Integrate both sides

Evaluate integrals

Exponentiate both sides

Write with single constant C = ±e^{C₁}

Example 3 — With initial conditionmedium

Solve , given .

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Separate (divide by y², multiply by dx)

Integrate left side

Integrate right side

Combine

Apply y(0) = 2

Particular solution

Example 4medium

Solve .

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Separate

Integrate left side

Integrate right side

General solution

Example 5hard

Solve .

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Rearrange

Separate (divide by tan y · (1-eˣ))

Integrate left side (substitute t = tan y)

Integrate right side (substitute u = 1-eˣ)

General solution

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