Sometimes it is possible to solve a nonlinear equation by...

Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear equation. The most important such equation has the form $$y' + p(t)y = q(t)y^n$$ and is called Bernoulli's equation after Jakob Bernoulli. If n≠ 0, 1, then the substitution $$v = y^{1-n}$$ reduces Bernoulli's equation to a linear equation. Solve the given Bernoulli equation by using this substitution. $$t^2y' + 2ty - y^3 = 0, t > 0$$ Ο $$y = \pm \frac{1}{\sqrt{\frac{1}{3t} + ct^2}}$$ Ο $$y = \pm \frac{2}{5t} + ct^4$$ Ο $$y = \pm \sqrt{\frac{2}{5t} + ct^4}$$ Ο $$y = \pm \frac{1}{\sqrt{\frac{2}{5t} + ct^4}}$$ Ο $$y = \pm \sqrt{\frac{1}{3t} + ct^2}$$

Transcript

00:01 Okay, so this is a bernoulli's differential equation.

00:04 Let's take the case when n is zero so let's solve the differential equation.

00:07 So when n is zero what happens is y dash plus p of t into y is equal to q of t into y power zero what is why is why is 0 which is one now this is a linear differential equation because it is of the form d y by d x plus a function of t into y is equal to a function of t for a linear differential equation we know pretty well that there is an integrating factor so what is the integrating factor if it's e power integral p of t d t and what is the solution of the differential equation so when you multiply this integrating factor both sides of the equation so i'll get y dash into e power integration of p of t d t plus p of t into y into e power integration of p of t d t is equal to q of t into e power integration of p of t d t so i multiply both sides but if you can see here this is an exact differential because this is something like a product rule so how do we write that so it is d by d t of so if you can remember product rule product rule d by d t of uv so let me write here d by d t of uv is u dv by d t plus v d u d u by d t so this is called as product rule so if you can see here this is of the form product rule so if i write d by d t of y into e power integration of p d d d d d d if you use product rule for this particular product so that means this is u this is v then you will get this part y dash into e power integral p of t d t plus y into what is differentiation of this one it is e power integration of p of t d t into by chain rule the derivative of the integral p of t which is p of t okay so now right side will be q of t into e power integration of t d d d d d t right now integrate both sides so when i integrate both sides this divide it goes off so it's y into e power integral p of t d t is equal to integration of q of t e power integration of p of t d t plus some constant of integration so this is the solution of the linear differential equation or a bernoulli differential equation when n is zero a simple linear differential equation because an is zero it is linear differential equation and we have used the method of integrating fact so for any linear differential equation please remember this is a formula y times integrating factor is integral of the right side function into integrating factor d t plus some constant of integration now let's see what happens to the bernowly equation when n is 1 so when n is 1 we have y dash plus q of t into y sorry p of t of t into y to right side so it's q of t minus p of t into y this is a variable separable differential equation i'll collect y's one side and t's the other side see this is a variable separable it's very easy to integrate this just integrate both sides so what is integration of d y by y it's l n y is equal to integration of q of t minus p of t i don't know what is this q of t p of t so i'll just keep it as it d t plus some construct of integration so since this is an ln, i'll take an exponential bosae.

03:32 So, y is equal to e -parc into e -power integration of q of t t minus p of t d t.

03:40 But since c is an arbitrary constant, e -par c is also some constant.

03:45 So it is k times e -power integration of q of t minus p of t d t.

03:51 This is the solution, the bernoulli equation when n is 1.

03:57 Now when n is other than 0 and 1.

03:59 So when it is a positive integer other than 0 and 1 not even a positive integer it can be anything basically so apart from 0 and 1 it can be any real number then what is our banally differential equation we have y -dash plus p of t into y is equal to q of t into y power n we use a change of variable v is equal to y power 1 minus n now let's differentiate both sides with respect to t so it is d v by d t is equal to 1 minus n into y power 1 minus n minus 1 into 3 minus 1 into 3...

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