Bernoulli Differential Equations AP Calculus BC Bernoulli Differential

Bernoulli Differential Equations AP Calculus BC

Bernoulli Differential Equations ØA Bernoulli differential equation can is of the form where P and Q are continuous functions on a given interval. ØNote that if n = 0 or 1, then we have a linear differential equation. ØBut what if n > 1? ØOur goal is to take this non-linear differential equation and turn it into a linear differential equation, so we can solve it.

Reduction § Divide by yn § We’re going to use u-substitution, so let u = y 1–n § So § Substitute back into original equation § Multiply by (1 – n) § The differential equation is now linear (ugly, but linear)!

Example 1 Find the general solution to the differential equation 1) n = 2, so u = y 1– 2, or u = y– 1. 2) Divide equation by y 2 to get RHS to equal x 3) Find du/dx 4) Solve for dy/dx

Example 1 (cont. ) 5) Substitute into equation in #2 6) Now this is a linear differential equation, so solve using integrating factor.

Example 1 (cont. ) 7) Multiply by x– 1 8) Product Rule in reverse 9) Integrate 10) Multiply by x 11) Substitute back for y (Recall u = y– 1) y– 1 = –x 2 + Cx

Example 2 Find the general solution to

Example 3 – Initial Value Problem Solve the differential equation y(1) = 0. with the initial condition

Still Confused? Watch this video: https: //www. youtube. com/watch? v=7 Mmhoqv. M 9_Q It helped me!