Question

The Bernoulli equation \(\frac{dy}{dx}+P(x)y=Q(x)y^3\) is best solved by making which of the following substitutions?

• A. \(v=y^{-2}\)
• B. \(v=y^{-1}\)
• C. \(v=y^2\)
• D. \(v=y^3\)

Hint:

Use the standard Bernoulli substitution \(v=y^{1-n}\) with \(n=3\).

Answer 1

The Correct Option is A

Step 1: The given equation \(\frac{dy}{dx}+P(x)y=Q(x)y^3\) is a Bernoulli equation with \(n=3\).

Step 2: Divide throughout by \(y^3\): \[y^{-3}\frac{dy}{dx}+P(x)y^{-2}=Q(x).\]

Step 3: Let \(v=y^{-2}\). Then \(\frac{dv}{dx}=-2y^{-3}\frac{dy}{dx}\), so \(y^{-3}\frac{dy}{dx}=-\frac{1}{2}\frac{dv}{dx}\).

Step 4: Substituting gives \[-\frac{1}{2}\frac{dv}{dx}+P(x)v=Q(x),\] which simplifies to the linear equation \(\frac{dv}{dx}-2P(x)v=-2Q(x)\).

Step 5: This confirms the standard Bernoulli rule \(v=y^{1-n}\) with \(n=3\) gives \(v=y^{-2}\).

\[\boxed{v=y^{-2}}\]