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!