Chapter 6.4, Problem 60E

### 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 . y ' + ( 1 x ) y = x y ,     x > 0

To determine

To calculate: The general solution of the provided Bernoulli equation y+(1x)y=xy.

Explanation

Given:

The Bernoulli differential equation,

yâ€²+(1x)y=xy â€¦â€¦ (1)

Calculation:

The general solution of the Bernoulli equation 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)

As equation (1) is of the form of Bernoulli equation,

yâ€²+P(x)y=Q(x)yn

Where, P(x)=1x,Q(x)=x,n=12

Therefore, solve y(1âˆ’n)eâˆ«(1âˆ’n)P(x)dx

y(1âˆ’n)eâˆ«(1âˆ’n)P(x)dx=y12eâˆ«(12)(1x)dx=y12e12lnx=y12x12=xy

Similarly solve âˆ«(1âˆ’n)Q(x)eâˆ«(1âˆ’n<

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