Find a general solution. Show the details of your calculation. 4y''+32y'+63y=0

Solve the given nonhomogeneous linear ODE by variation of parameters or undetermined coefficients. Show the details of your work.

$$(D^2-4D+4l)y=6e^{2x}/x^4$$

Find the Wronskian. Show linear independence by using quotients and confirm it. cosh ax, sinh ax

Find a (real) general solution. State which rule you are using. Show each step of your work.

$$(D^2+2D+\dfrac{3}{4}l)y=3e^x+\dfrac{9}{2}x$$

Suppose the population $P$ of a certain species of fish depends on the number $x$ (in hundreds) of a smaller fish that serves as its food supply, so that

$$P(x)=2 x^{2}+1.$$

Suppose, also, that the number of the smaller species of fish depends on the amount a (in appropriate units) of its food supply, a kind of plankton. Specifically, $x=f(a)=3 a+2$. A biologist wants to find the relationship between the population $P$ of the large fish and the amount a of plankton available, that is,$P[f(a)]$. What is the relationship?

Suppose the population P of a certain species of fish depends on the number x (in hundreds) of a smaller kind of fish that serves as its food supply, where $P(x)=2 x^{2}+1$. Suppose, also, that the number x (in hundreds) of the smaller species of fish depends on the amount a (in appropriate units) of its food supply, a kind of plankton, where x=f(a)=3a+2. Find $(P \cdot f)(a)$, the relationship between the population P of the large fish and the amount a of plankton available.

If a wet sheet in a dryer loses its moisture at a rate proportional to its moisture content, and if it loses half of its moisture during the first 10 min of drying, when will it be practically dry, say, when will it have lost 99% of its moisture? First guess, then calculate.

Assume that you fish for 3 years, then fishing is banned for the next 3 years. Thereafter you start again. And so on. This is called intermittent harvesting. Describe qualitatively how the population will develop if intermitting is continued periodically. Find and graph the solution for the first 9 years, assuming that A=B=1, H=0.2, and y(0)=2.

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. y'-0.4y=29 sin x

Find a real general solution. Show the details of your work. (x²D²-4xD+6l)y=C

(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.

$$4x^2y''-3y=0, y(1)=-3, y(1)=0, x^{3/2},x^{-1/2}$$

(a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. (b) Solve the IVP. Graph or sketch the solution.

$$\text{y''+0.6y'+0.09y=0, y(0)=2.2, y'(0)=0.14, }e^{-0.3x}\text{, }xe^{-0.3x}$$

In tuning a stereo system to a radio station, we adjust the tuning control (turn a knob) that changes C (or perhaps L) in an RLC-circuit so that the amplitude of the steady-state current becomes maximum. For what C will this happen?

Find a real general solution. Show the details of your work. x²y''+0.7xy'-0.1y=0