91Ó°ÊÓ

Determine whether the differential equation is linear or nonlinear. $$\frac{d^{2} y}{d x^{2}}+e^{x} \frac{d y}{d x}=x^{2}$$.

Determine the order of the differential equation. $$\frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=e^{x}$$

A tank initially contains 600 L of solution in which there is dissolved 1500 grams of chemical. A solution containing \(5 \mathrm{g} / \mathrm{L}\) of the chemical flows into the tank at a rate of \(6 \mathrm{L} / \mathrm{min},\) and the well-stirred mixture flows out at a rate of 3 L/min. Determine the concentration of chemical in the tank after one hour.

Determine whether the given function is homogeneous of degree zero. Rewrite those that are as functions of the single variable \(V=y / x\). $$f(x, y)=\frac{5 x+2 y}{9 x-4 y}$$

Determine the differential equation giving the slope of the tangent line at the point \((x, y)\) for the given family of curves. $$y=c e^{2 x}$$

Solve the given differential equation. $$y^{\prime \prime}-2 y^{\prime}=6 e^{3 x}$$

A boy 2 meters tall shoots a toy rocket straight up from head level at 10 meters per second. Assume the acceleration of gravity is 9.8 meters/sec \(^{2}\). (a) What is the highest point above the ground reached by the rocket? (b) When does the rocket hit the ground?

Determine whether the given differential equation is exact. $$y e^{x y} d x+\left(2 y-x e^{x y}\right) d y=0$$

Solve the given differential equation. $$\frac{d y}{d x}+y=4 e^{x}$$

The number of bacteria in a culture grows at a rate that is proportional to the number present. Initially there were 10 bacteria in the culture. If the doubling time of the culture is 3 hours, find the number of bacteria that were present after 24 hours.