Ch 07/Exact Differential Equations/H.K. Dass 3.11-3.12

Exact Differential Equations

M dx + N dy = 0 is exact if dM/dy = dN/dx. The solution: integrate M w.r.t. x (y constant) plus the y-only terms of N integrated w.r.t. y.

IExactness and Solution

is exact if .

How to Solve:

1. Verify: .

2. Integrate w.r.t. (y constant) .

3. From , identify terms without . Integrate w.r.t. .

4. Solution: .

If not exact, multiply by an I.F.:

Rule 1: If , then I.F. .

Rule 2: If , then I.F. .

Rule 3: If , , then I.F. .

Rule 5: If homogeneous and , then I.F. .

Example 1 (Exact)easy

Solve .

Show step-by-step solution+

Verify

. Exact.

Integrate M w.r.t. x

N terms without x

Solution

Example 2 (Using Rule 5 I.F.)hard

Solve .

Show step-by-step solution+

Homogeneous, degree 3

. I.F. .

Multiply and verify exact

. Now exact.

Solution

← Previous

Ch 6 · Bernoulli Equations

Next chapter →

Ch 8 · Second Order Linear DEs with Constant Coefficients