Bernoulli equation

Bernoulli Differential Equations Differential equation in the form ddxy p(x) y  q(x)yn where p(x) and q(x) are continuous functions on the interval we’re working on and n is a real number. Differential equations in this form are called Bernoulli Equations.

Bernoulli equation in the form ddxy  p(x) y  q(x)yn First notice that if n = 0 or n = 1 then the equation is linear Therefore, in this section we’re going to be looking at solutions for values of n other than these two.

Steps in solving the equation are as follows: ddxy p(x) y  q(x)yn 1. Divide the differential equation by yn to get, yndy + p(x)y1ndx = q(x)dx 2. Substitution z = y1n to convert this into a differential equation in terms of z. As we’ll see this will lead to a differential equation that we can solve. 3. Find dz to get, dz = (1- n)yndy 1 1 n dz = y  n dy  4. Plugging in step 1 to gives, 1 1 n dz + p(x)y1n dx = q(x)dx  This is a linear differential equation that we can solve for v and once we have this in hand we can also get the solution to the original differential equation by plugging z back into our substitution and solving for y.

Example 20 Solve Bernoulli Differential Equation. dy + y = x dx 2x y3



Do Activities ACTIVITY EXERCISES 5 (Bernoulli Equation) Solve Bernoulli Equation dy  2xy  xy 4 dx Upload files and must be submit it to Google Classroom within 5th November 2020 no later than 3 PM.