Ch 02/Variables Separable Method/H.K. Dass 3.6

Variables Separable Method

If you can rearrange a first-order DE so that all y terms (including dy) are on one side and all x terms (including dx) are on the other, just integrate both sides.

IWhat is this method?

If you can rearrange a first-order DE so that:

- ALL the stuff (including ) is on one side

- ALL the stuff (including ) is on the other side


then you have "separated the variables." Just integrate both sides.


Working Rule:

1. Rearrange:

2. Integrate both sides:

3. Add constant on one side.

Note

and are standard formulae you should memorize.

IIDirect Separation and Substitution Type

Sometimes variables cannot be separated directly. Common substitutions:

- If appears: put

- If appears: put


After substitution, the new equation in becomes separable.

Example 1 (Easy)easy

Solve .

Show step-by-step solution+
Separate variables
Integrate both sides
Final answer
Example 2 (Medium)medium

Solve .

Show step-by-step solution+
Separate
Integrate left side
Using integration by parts on : result is .
Integrate right side
The terms cancel beautifully to give .
Final answer
Example 3 (Substitution)medium

Solve .

Show step-by-step solution+
Rewrite
. Not directly separable.
Substitute z = x + y
.
Separate and integrate
Using half-angle identities: .
Substitute back
Example 4 (Hard)hard

Solve .

Show step-by-step solution+
Rearrange ratio
Apply componendo-dividendo
Recognize exact differentials and .
Integrate
Final answer
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