Cs130 day11

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

CS 130: Mathematical Methods in Computer Science Ordinary Differential Equations Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph Day 11

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Ordinary Differential Equations

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Ordinary Differential Equations

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Bernoulli’s Equation y 0 + p(x)y = q(x)y n

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Bernoulli’s Equation y 0 + p(x)y = q(x)y n

The Bernoulli equation may be transformed into a linear differential equation through a change of variables.

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Bernoulli’s Equation y 0 + p(x)y = q(x)y n

The Bernoulli equation may be transformed into a linear differential equation through a change of variables.

Example: 1. y 0 + 2y = xy 3

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Bernoulli’s Equation y 0 + p(x)y = q(x)y n

The Bernoulli equation may be transformed into a linear differential equation through a change of variables.

Example: 1. y 0 + 2y = xy 3 dy 2. x dx + y = x2 y 2

3. y 0 + y cot x = y 3 sin x

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Ordinary Differential Equations

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Reduction of order

Consider the second-order equation y 00 + p(x)y 0 = f (x).

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Reduction of order

Consider the second-order equation y 00 + p(x)y 0 = f (x).

Use the substitution u = y 0 to convert the equation into a new first-order ODE involving the function u.

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Reduction of order

Consider the second-order equation y 00 + p(x)y 0 = f (x).

Use the substitution u = y 0 to convert the equation into a new first-order ODE involving the function u.

Example: Use reduction of order to solve each of the following second-order IVPs. 1. y 00 + 2y 0 = 4 where y(0) = 2, y 0 (0) = 1 2. y 00 +

1 0 4−x y

= 4 − x where y(0) = 1, y 0 (0) = 1

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Reduction of order

Consider the second-order equation y 00 = g(y 0 )h(x).

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Reduction of order

Consider the second-order equation y 00 = g(y 0 )h(x).

Use the substitution u = y 0 to convert the equation into a new first-order ODE involving the function u.

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Reduction of order

Consider the second-order equation y 00 = g(y 0 )h(x).

Use the substitution u = y 0 to convert the equation into a new first-order ODE involving the function u.

Example: Use reduction of order to solve each of the following second-order IVPs. √ 1. y 00 = y 0 where y(0) = 3, y 0 (0) = 4 2. (y 0 )2 y 00 = x2 where y(0) = 1, y 0 (0) = 0

Bernoulli’s Equation

Converting certain second-order ODEs to first-order ODEs

Questions?

See you next meeting!