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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