Chapter 6.4, Problem 61E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Solving a Bernoulli Differential Equation In Exercises 57-64, solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the form y ' + P ( x ) y = Q ( x ) y n that can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is y 1 − n e f ( 1 − n ) P ( x ) d x   = ∫ ( 1 − n ) Q ( x ) e f ( 1 − n ) P ( x ) d x d x + C . x y ' + y = x y 3 ,     x > 0

To determine

To calculate: The general solution of the Bernoulli equation xy+y=xy3 granted that the general solution of the Bernoulli equation y+P(x)y=Q(x)yn is y(1n)e(1n)P(x)dx=(1n)Q(x)e(1n)P(x)dxdx+C

Explanation

Given:

The Bernoulli differential equation,

xyâ€²+y=xy3 â€¦â€¦ (1)

The solution of Bernoulli equation yâ€²+P(x)y=Q(x)yn is

y(1âˆ’n)eâˆ«(1âˆ’n)P(x)dx=âˆ«(1âˆ’n)Q(x)eâˆ«(1âˆ’n)P(x)dxdx+C

Formula used:

The integration of:

âˆ«eaxdx=eaxa+C

The solution of Bernoulli equation yâ€²+P(x)y=Q(x)yn is

y(1âˆ’n)eâˆ«(1âˆ’n)P(x)dx=âˆ«(1âˆ’n)Q(x)eâˆ«(1âˆ’n)P(x)dxdx+C â€¦â€¦ (2)

Calculation:

Consider the equation xyâ€²+y=xy3

Since it is a Bernoulli equation so

P(x)=1x,Q(x)=1,n=3

The solution of Bernoulli equation yâ€²+P(x)y=Q(x)yn is

y(1âˆ’n)eâˆ«(1âˆ’n)P(x)dx=âˆ«(1âˆ’n)Q(x)eâˆ«(1âˆ’n)P(x)dxdx+C

For y(1âˆ’n)e

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