Ch 08/Exact Differential Equations/H.K. Dass — Section 3.11, p.154

Exact Differential Equations

A DE M dx + N dy = 0 is exact if ∂M/∂y = ∂N/∂x. Solution found by direct integration.

IExactness Condition & Working Rule

A DE written as is exact if there exists a function such that:

This means and .

Condition for exactness (H.K. Dass §3.11):

Working Rule:

1. Check the exactness condition

2. Integrate M w.r.t. x (keeping y constant) → call it

3. Identify terms in N that are not in → integrate those w.r.t. y → call it

4. Solution:

Note

The solution is just . Step 2 recovers the x-part of F, and step 3 picks up any y-only terms that Step 2 missed.

Example 1easy

Solve .

Show step-by-step solution+

Identify M and N

Check exactness

, ✓ Exact.

Integrate M w.r.t. x (y constant)

Differentiate f w.r.t. y

Find missing N terms: N - ∂f/∂y = 2y - x... wait

Compare with N = x+2y: the x part is captured, missing = 2y

Integrate missing term w.r.t. y

Solution

Example 2 (H.K. Dass Ex.21)medium

Solve .

Show step-by-step solution+

Check ∂M/∂y

Check ∂N/∂x

✓ Exact.

Integrate M w.r.t. x (y constant)

Compare ∂f/∂y with N

; N has extra

Integrate -5y⁴ w.r.t. y

Solution

Example 3hard

Solve .

Show step-by-step solution+

M = y/x + 6x, N = ln x - 2

Check ∂M/∂y = 1/x

Check ∂N/∂x = 1/x

✓ Exact.

Integrate M w.r.t. x

Find ∂f/∂y and compare with N

; N = ; missing:

Integrate -2 w.r.t. y

Solution

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