91Ó°ÊÓ

give an example of: A differential equation that is not separable.

Give an example of: Values of the spring constant \(k\), the mass \(m,\) and the damping coefficient \(a\) so that the motion is underdamped and shows damped oscillations.

give an example of: A differential equation all of whose solutions form the family of functions \(f(x)=x^{2}+C\)

give an example of: A differential equation all of whose solutions form the family of hyperbolas \(x^{2}-y^{2}=C\)

Are the statements true or false? Give an explanation for your answer. A differential equation of the form \(d y / d x=f(x)\) is separable.

Are the statements true or false? Give an explanation for your answer. A differential equation of the form \(d y / d x=1 / g(y)\) is separable.

Are the statements true or false? Give an explanation for your answer. For all constants \(k\), the equation \(y^{\prime}+k y=0\) has exponential functions as solutions.

Are the statements true or false? Give an explanation for your answer. The differential equation \(d y / d x=x+y\) can be solved by separation of variables.

Are the statements true or false? Give an explanation for your answer. The differential equation \(d y / d x-x y=x\) can be solved by separation of variables.

Are the statements true or false? Give an explanation for your answer. The only solution to the differential equation \(d y / d x=\) \(3 y^{2 / 3}\) passing through the point (0,0) is \(y=x^{3}\)