Bernoulli Equations

Author: John J Weber III, PhD Corresponding Textbook Sections:

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Bernoulli Equations

Definition: Bernoulli Equations

A first-order equation that can be written in the form

dydx+P(x)y=Q(x)yn,

where P(x) and Q(x) are continuous on an interval (a,b) and n is a real number, is called a Bernoulli equation.

Method for Solving Bernoulli Equations

  1. Use the substitution z=y1n.

  2. Differentiate using chain rule: dzdx=(1n)yndydx.

  3. Multiply both sides of the equation by (1n)yn results in (1n)yndydx+(1n)P(x)y1n=(1n)Q(x).

  4. Rewrite the DE as dzdx+(1n)P(x)z=(1n)Q(x) which is first-order linear.

  5. Resubstitute y1n for z.

  6. Identify missing solutions or solutions that are not solutions of the original DE.

Question 03

Solve the following DEs:

  1. dydx+2xy=e3x2+2xy4

  2. dydx=y4cos(x)+ytan(x)

  3. xdydx+y=y2x

  4. xdydxy=ex2y5

  5. y1xy=12y

  6. y+x1x2y=xy, y(0)=1

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Last Modified: Sunday, 6 September 2020 14:15 EDT