Does the initial value problem (x-2)y'=y, y(2)=1 have a solution? Does your result contradict our present theorems?

Common stock value-Constant growth McCracken Roofing, Inc., common stock paid a dividend of $\$ 1.20$ per share last year. The company expects earnings and dividends to grow at a rate of $5 \%$ per year for the foreseeable future. Find the required rate of return for this stock that would result in a price per share of $\$ 28$.

Find a general solution. Check your answer by substitution ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9. 4y''-25y=0

Calculate the double integral through R xy^2 /x^2+1 dA, R={(x,y) | 0<=x<=2, 1<=y<=2}

Can two solution curves of the same ODE have a common point in a rectangle in which the assumptions of the present theorems are satisfied?

It can be shown that the curve y(x) of an inextensible flexible homogeneous cable hanging between two fixed points is obtained by solving y''=k√1+y'², where the constant k depends on the weight. This curve is called catenary (from Latin catena = the chain). Find and graph y(x), assuming that k=1 and those fixed points (-1,0) and (1,0) in a vertical xy-plane.

Find the curve through the origin in the xy-plane which satisfies y''=2y' and whose tangent at the origin has slope 1.

The tank contains 80 lb of salt dissolved in 500 gal of water. The inflow per minute is 20 lb of salt dissolved in 20 gal of water. The outflow is 20 gal min of the uniform mixture. Find the time when the salt content y(t) in the tank reaches 95% of its limiting value (as t→∞).

On May 3, Zirbal Corporation purchased 4,000 shares of its own stock for $\$ 36,000$ cash. On November 4 , Zirbal reissued 850 shares of this treasury stock for $\$ 8,500$. Prepare the May 3 and November 4 journal entries to record Zirbal's purchase and reissuance of treasury stock.

On May 3, Zirbal Corporation purchased 4,000 shares of its own stock for $36,000 cash. On November 4, Zirbal reissued 850 shares of this treasury stock for$8,500. Prepare the May 3 and November 4 journal entries to record Zirbal’s purchase and reissuance of treasury stock.

Let the isotherms (curves of constant temperature) in a body in the upper half-plane y>0 be given by 4x²+9y²=c. Find the orthogonal trajectories (the curves along which heat will flow in regions filled with heat-conducting material and free of heat sources or heat sinks).

If there is a heat transfer from the lateral surface of a thin wire of length L into a medium at constant temperature $\boldsymbol{u}_{m}$, then the heat equation takes on the form $k \frac{\partial^{2} u}{\partial x^{2}}-h\left(u-u_{m}\right)=\frac{\partial u}{\partial t}, \quad 0<x<L, t>0$, where h is a constant. Find the temperature u(x, t) if the initial temperature is a constant $u_{0}$ throughout and the ends x = 0 and x = L are insulated.

Heating and cooling of a building can be modeled by the ODE T'=k1(T-Ta)+k(T-Tω)+P, where T=T(t) is the temperature in the building at time t, Ta the outside temperature, Tω the temperature wanted in the building, and P the rate of increase of T due to machines and people in the building, and k1 and k2 are (negative) constants. Solve this ODE, assuming P=const, Tω=const, and Ta varying sinusoidal over 24 hours, say, Ta=A-C cos(2π/24)t. Discuss the effect of each term of the equation on the solution.

If p and r in y'+p(x)y=r(x) are continuous for all x in an interval |x-x0|≤a, show that f(x, y) in this ODE satisfies the conditions of our present theorems, so that a corresponding initial value problem has a unique solution. Do you actually need these theorems for this ODE?