Respuesta :
Answer:
a) [tex]y = - \frac{1}{x+c}[/tex]
b) y = 0
Step-by-step explanation:
Solution:-
- We are given a ODE of the form:
[tex]\frac{dy}{dx} = y^2[/tex]
- To solve the given ODE and determine the general solution we will separate the variables in the given ODE as follows:
[tex]\frac{dy}{y^2} = dx[/tex]
- Integrate both sides and determine a general explicit function of (y):
[tex]\int {y^-^2} \, dy = \int {} \, dx + c\\\\-\frac{1}{y} = x + c\\\\y = -\frac{1}{x+c}[/tex]
b) The singular solution that exist but is not included is the trivial solution corresponding to y = 0. This solution satisfies the the given ODE
We want to solve the given differential equation, we will find:
- a) [tex]y = \frac{1}{(k - x)} [/tex]
- b) y = 0
- c) (a, 1/(9/4 - a))
How to solve the differential equation?
So we have:
[tex]\frac{dy}{dx} = y^2[/tex]
a) if we define:
[tex]y = \frac{1}{(k - x)} [/tex]
And we derivate that, we will get:
[tex]y' = \frac{1}{(k - x)^2} = y^2[/tex]
So that is the general solution.
b) We would want a singular solution that is not included in the general form. Here we could use something like:
y = 0 we get:
dy/dx = 0 = 0^2
This is a trivial singular solution.
c) If we have y(a) = b, then:
[tex]y = \frac{1}{(k - a)} = b[/tex]
To find the value of k we use the other condition:
y'(2) = 4*y(2)
[tex]\frac{1}{(k-2)^2} = 4*\frac{1}{k - 2} \\ \\ \frac{1}{k - 2} = 4\\\\ (k - 2) = 1/4\\\\ k = 2 + 1/4 = 9/4[/tex]
Then we have:
[tex]y = \frac{1}{(9/4 - a)} = b[/tex]
So the points (a, b) are of the form: (a, 1/(9/4 - a))
If you want to learn more about differential equations you can read:
https://brainly.com/question/353770
