Ch 07/Bernoulli's Equation/H.K. Dass — Section 3.10, p.150

Bernoulli's Equation

Bernoulli form: dy/dx + Py = Qyⁿ. Substitute z = y^(1-n) to linearize.

IBernoulli Form & Reduction

Standard Bernoulli form:

This is non-linear due to . The trick is a substitution that converts it to linear.

Reduction (H.K. Dass §3.10):

1. Divide both sides by :

2. Let , so

3. Substitute: DE becomes linear in z:

4. Solve using I.F. method, then back-substitute

Note

When n = 0, the Bernoulli equation is already linear (just the standard linear DE). When n = 1, divide by y to get a separable equation. The Bernoulli substitution is only needed for n ≠ 0, 1.

Example 1medium

Solve .

Show step-by-step solution+

Identify n = 2

P = -1/x, Q = -1/x², n = 2

Divide by y²

Let z = y^{1-2} = y^{-1} = 1/y

, so multiplying by -1:

Linear DE in z

I.F. = x (P = 1/x)

Integrate

Back-substitute z = 1/y

Example 2 (H.K. Dass Ex.13)hard

Solve .

Show step-by-step solution+

Divide by x² dx

Bernoulli with n=2, P=1/x, Q=-1/x²

Divide by y²

z = 1/y, dz/dx = -y⁻²y'

I.F. = e^{-∫1/x dx} = 1/x

Integrate

Back-substitute z = 1/y

Example 3 — Log substitution (H.K. Dass Ex.14)hard

Solve .

Show step-by-step solution+

Divide by xy

Let z = log y → dz/dx = y'/y

Standard form

I.F. = 1/x

Integrate (by parts)

General solution

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