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II.

2 y^{\prime}=-(1 / x) y+\pi x y^{2}

, Resolviendo la ecuación justificar qu...

Feb 25, 2024

II.

2 y^{\prime}=-(1 / x) y+\pi x y^{2}

, Resolviendo la ecuación justificar que la familia uniparamétrica de soluciones es

y(x)=1 /\left(C x-\pi x^{2}\right), \quad C \in \mathbf{R}

. Cuando hay otra solución que no está dentro de la familia, es decir, no se puede obtener al dar valores a la constante

C \in \mathbf{R}

, dicha solución se denomina singular. ¿Es solución singular la solución

y(x) \equiv 0

(compruébese que es solución)?

Solution by Steps

step 1

To solve the differential equation

2 y' = -\frac{1}{x} y + \pi x y^2

, we look for a uniparametric family of solutions

step 2

We propose a solution of the form

y(x) = \frac{1}{Cx - \pi x^2}

, where

C

is a parameter

step 3

To verify that

y(x)

is a solution, we calculate its derivative

y'(x)

step 4

The derivative is

y'(x) = \frac{C - 2\pi x}{(Cx - \pi x^2)^2}

step 5

We substitute

y(x)

and

y'(x)

into the original differential equation to check if it holds

step 6

After substitution, we find that the equation is satisfied, confirming that

y(x)

is indeed a solution

step 7

To determine if

y(x) \equiv 0

is a singular solution, we substitute

y(x) = 0

into the differential equation

step 8

We find that

0 = 0

, which means

y(x) \equiv 0

is indeed a solution to the differential equation

step 9

Since

y(x) \equiv 0

cannot be obtained from the uniparametric family by assigning a value to

C

, it is a singular solution

Answer

The uniparametric family of solutions is

y(x) = \frac{1}{Cx - \pi x^2}

, and

y(x) \equiv 0

is a singular solution.

Key Concept

Uniparametric family of solutions and singular solutions in differential equations

Explanation

A uniparametric family of solutions is a set of solutions to a differential equation that can be obtained by varying a parameter. A singular solution is a solution that cannot be derived from the family by any choice of the parameter. In this case,

y(x) \equiv 0

is a singular solution because it cannot be obtained from the family

y(x) = \frac{1}{Cx - \pi x^2}

for any value of

C