38 9. Problems In each of Problems 1 through 8, solve the given differential equation. 1. y' = CHAPTER 2 First-Order Differential Equations 2. y' + y² sin.x = 0 3. y'= cos(x) cos² (2y) 4. xy' = (1 - y2) 1/2 dy 5. x-e-x y + ey 2 dx dy dx = 1+ y2 y dx x dy dx dy 7. = -X 8. y In each of Problems 9 through 16: a. Find the solution of the given initial value problem in explicit form. Gb. Plot the graph of the solution. c. Determine (at least approximately) the interval in which the solution is defined. 581 Some of the results requested in Problems 17 through 22 can be obtained either by solving the given equations analytically or by plotting numerically generated approximations to the solutions. Try to form an opinion about the advantages and disadvantages of each approach. G 17. Solve the initial value problem y'=(1-2x) y2, y(0) = -1/6 10. y'=(1-2x)/y, y(1) = -2 11. xdx+ye *dy = 0, y(0) = 1 12. dr/de = r²/0, r(1) = 2 13. y'=xy³(1+x²)-1/2, y(0) = 1 14. y' = 2x/(1+2y), y(2) = 0 albedingulos a aloc (ut.) i susood bnf 23. 15. y'= (3x² - e*)/(2y-5), y(0) = 1 16. sin(2x) dx + cos(3y) dy = 0, y(π/2) = π/3 1+3x² 3y² - 6y' and determine the interval in which the solution is valid. Hint: To find the interval of definition, look for points where the integral curve has a vertical tangent. G 18. Solve the initial value problem 3x² 3y² - 4' y' = y(0) = 1 y(1) = 0 G 19. Solve the initial value problem y' = 2y² + xy², y(0) = 1 and determine where the solution attains its minimum value, G20. Solve the initial value problem y' = 2-et 3+2y' y(0) = 0 and determine where the solution attains its maximum value. G21. Consider the initial value problem y' = y(0) = yo. and determine the interval in which the solution is valid. Hint: To find the interval of definition, look for points where the integral curve has a vertical tangent. sidEnsink oom G22. a. Determine how the behavior of the solution as t increases depends on the initial value yo. b. Suppose that yo= 0.5. Find the time T at which the solution first reaches the value 3.98. ty(4-y) 3 Consider the initial value problem y' = a. Determine how the solution behaves as t→ ∞0. b. If yo2, find the time T at which the solution first reaches the value 3.99. c. Find the range of initial values for which the solution lies in the interval 3.99< y < 4.01 by the time t = 2. Solve the equation tanti insienody dou? ay + b cy + d do sw (S) bas (OS) snomus ty(4-y) 1+t dQ dt y(0) = yo > 0. dx where a, b, c, and d are constants. 24. Use separation of variables to solve the differential equation =r(a+bQ), Q(0) = 20, 101 (5) mus where a, b, r, and Qo are constants. Determine how the solution behaves as t → ∞ ban dhe mbungsi gel Homogeneous Equations. If the right-hand side of the equation dy/dx = f(x, y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous. Such equations can always be transformed into separable equations by a change of the dependent variable. Problem 25 illustrates how to solve first-order homogeneous equations. hoo 7d nat The word "homogeneous" has different meanings in different mathematical contexts. The homogeneous equations considered here have nothing to do with the homogeneous equations that will occur in Chapter 3 and elsewhere. N