High School

7. Write any example of a homogenesis of degree aro ODE that is non-linear Write any example of a homogeneous linear ODE. Make sure you know the differences between the two,

8. Write in example (it's possible) of an ODE that is both Bernoulli AND homogeneous of degree zero.

Answer :

1. A common example of a non-linear homogeneous ordinary differential equation (ODE) is: [tex]\[y' = \frac{x^2}{y}\][/tex] and for homogeneous linear ODE is:

[tex]\[y' + 2xy = 0\][/tex].

2. An ODE that is both Bernoulli and homogeneous of degree zero is:

[tex]\[y' - \frac{2x}{x^2 + y^2}y = y^3\][/tex]

1. A common example of a non-linear homogeneous ordinary differential equation (ODE) is:

[tex]\[y' = \frac{x^2}{y}\][/tex]

This equation is non-linear because of the division involving [tex]\(x^2\)[/tex] and y, and it's homogeneous of degree zero because the sum of the exponents of x and y in the equation (0 - 1 = -1) is equal to zero.

2. A common example of a homogeneous linear ODE is:

[tex]\[y' + 2xy = 0\][/tex]

This equation is linear because it contains no products of y and its derivatives, and it's homogeneous because it's of the first degree in y and its derivatives.

3. An example of an ODE that is both Bernoulli and homogeneous of degree zero is:

[tex]\[y' - \frac{2x}{x^2 + y^2}y = y^3\][/tex]

This equation is Bernoulli because it can be transformed into a linear ODE through the substitution [tex]\(z = y^{1 - n}\)[/tex], where n = 3 in this case.

After substitution, it becomes a linear homogeneous ODE, and it's also homogeneous of degree zero because the sum of the exponents of x and y in the equation [tex]\(-\frac{2x}{x^2 + y^2}\)[/tex] is equal to zero.

Learn more about homogeneous equation here:

https://brainly.com/question/34858104

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