Question

16. y = 0 (that is, y(x) = 0 for all x, also written y(x) = 0) is a solution of (2) (not of (1) if (x) • o , called the trivial solution 17. The sum of a solution of (1) and a solution of (2) is a solution of (1). 18. The difference of two solutions of (1) is a solution of (2). 19. If yı is a solution of (1), what can you say about cy? 20. If y, and yz are solutions of yí + pyn = 1 and ya + py2 = 1, respectively with the same p!), what can you say about the sum yı + y2? 21. Variation of parameter. Another method of obtaining (4) results from the following idea Write (3) as cy*, where y* is the exponential function, which is a solution of the homogeneous linear ODE y*' + py* = 0. Replace the arbitrary constant c in (3) with a function u to be determined so that the resulting function y = uy* is a solution of the nonhomogeneous linear ODE y + 2y = 1. (b) Show that y = Y=x is a solution of the ODE y' - (2x + 1) y = -x - x - x + land solve this Riccati equation, showing the details. © Solve the Clairaut equation y2 - xy + y = 0 as follows. Differentiate it with respect to x, obtaining y" (2y' - x) = 0. Then solve (A) y" = 0 and (B) 2y' - x = 0 separately and substitute the two solutions (a) and (b) of (A) and (B) into the given ODE. Thus obtain (a) a general solution (straight lines) and (b) a parabola for which those lines (a) are tangents (Fig. 6 in Prob. Set 1.1); so (b) is the envelope of (a). Such a solution (b) that cannot be obtained from a general solution is called a singular solution. (d) Show that the Clairaut equation (15) has as solutions a family of straight lines y = cx + g(c) and a singular solution determined by g'(s) = -x, where s = y', that forms the envelope of that family. 31-40 MODELING. FURTHER APPLICATIONS 31. Newton's law of cooling. If the temperature of a cake is 300°F when it leaves the oven and is 200°F ten minutes later, when will it be practically equal to the room temperature of 60°F, say, when will it be 61 F? 32 Heating and cooling of a building. Heating and cooling of a building can be modeled by the ODE T' = k (T-1) + kg(T-T) + P, 22-28 NONLINEAR ODES Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it. 22. y' + y = y2 wo)= -- 23. y' + xy = xy-!, y(0) = 3 24. y' + y = -x/y 25. y' = 3.2y - 10ya 26. y' = (tan y)/(x - 1), WO) = 5 27. y' = 1/6 - 2x) 28. 2.yy' + (x - 1)2 = xe (Set y2 = z) 29. REPORT PROJECT. Transformation of ODES. We have transformed ODEs to separable form, to exact form, and to linear form. The purpose of such transformations is an extension of solution methods to larger classes of ODES. Describe the key idea of each of these transformations and give three typical exam- ples of your choice for each transformation. Show each step (not just the transformed ODE). 30. TEAM PROJECT. Riccati Equation. Clairaut Equation. Singular Solution. A Riccati equation is of the form (14) y' + p(xy = g() + hợ). A Clairaut equation is of the form (15) y = xy' + 80'). (a) Apply the transformation y = y + 1/u to the Riccati equation (14), where Y is a solution of (14), and obtain for u the linear ODE ' + (2Yg - p)u = -8. Explain the effect of the transformation by writing it as y = Y + v, v = 1/u. where T = T() is the temperature in the building at time t, T, the outside temperature, T, the temperature wanted in the building, and P the rate of increase of T due to machines and people in the building, and k, and ka are (negative) constants. Solve this ODE, assuming P = const, Toconst, and T, varying sinusoidally over 24 hours, say, T. = A - C cos(27/24). Discuss the effect of each term of the equation on the solution. 33. Drug injection. Find and solve the model for drug injection into the bloodstream if, beginning at t = 0, a constant amount A 8/min is injected and the drug is simultaneously removed at a rate proportional to the amount of the drug present at time t. 34. Epidemics. A model for the spread of contagious diseases is obtained by assuming that the rate of spread is proportional to the number of contacts between infected and noninfected persons, who are assumed to move freely among each other. Set up the model. Find the equilibrium solutions and indicate their stability or instability. Solve the ODE. Find the limit of the proportion of infected persons as → and explain what it means. 35. Lake Erle. Lake Erie has a water volume of about 450 km and a flow rate (in and out) of about 175 km2

I need help with question 30d

Answer 1

30d) The Clairaut Equation is $y=xy'+g(y')$ --------------(1)

Let $s=y'$ ------------(2) be a parameter than (1) becomes

$y=xs+g(s)$

Differentiate with respect to x gives

$y'=x{ds\over dx}+s(1)+g'(s){ds\over dx} \\ \Rightarrow s=s+[x+g'(s)]{ds\over dx} ~~~~~~~~~~~~\text{ since }s=y' \\ \Rightarrow [x+g'(s)]{ds\over dx} =0 \\ \Rightarrow x+g'(s) =0~~\text{ or }~~~{ds\over dx}=0$

Now, ${ds\over dx}=0$ gives $s=c$ where c is arbitrary constant

Then we get from (1) and (2)

$y=cx+g(c)$ as the general solution ( which is family of straight lines)
Also $x+g'(s) =0 ~,~\Rightarrow g'(s)=-x$ gives the singular solution.