9 In each of Problems 1 through 8, solve the given differential equation.x²yy' + y² sinx = 0y' = cos²(x) cos² (2y)xy' = (1 - y²) 1/2dyx-e-xdxy + eyx²1+ y²1.2.3.4.5.6.7.y' =dydxdydxdydx==باد8.In each of Problems 9 through 16:form.a. Find the solution of the given initial value problem in explicitGb. Plot the graph of the solution.c. Determine (at least approximately) the interval in which thesolution is defined.9. y' = (1-2x) y²,10. y' = (1-2x)/y,11. xdx+ye-*dy = 0, y(0) = 112. dr/d0 = r²/0, r(1) = 213. y'=xy³ (1+x²)-1/2, y(0) = 114. y' = 2x/(1+2y), y(2) = 015. y' = (3x² - e*)/(2y-5), y(0) = 116. sin(2x) dx + cos(3y) dy = 0, y(π/2) = π/3Some of the results requested in Problems 17 through 22 can beobtained either by solving the given equations analytically or byplotting numerically generated approximations to the solutions. Tryto form an opinion about the advantages and disadvantages of eachapproach.G 17. Solve the initial value problemy'y(0) = -1/6y(1) = -2=y' ==1+3x²Зу2 — буand determine the interval in which the solution is valid.Hint: To find the interval of definition, look for points where theintegral curve has a vertical tangent.G 18. Solve the initial value problemy(0) = 13x²3y² - 4'y(1) = 0and determine the interval in which the solution is valid.Hint: To find the interval of definition, look for points where theintegral curve has a vertical tangent.G 19. Solve the initial value problemand determine where the solution attains its minimum value.G 20. Solve the initial value problem2- et3+2yy' = 2y² + xy², y(0) = 1y(0) = 0and determine where the solution attains its maximum value.G 21. Consider the initial value problemG 22.y' =y' =dQdt=ty(4- y)3a. Determine how the behavior of the solution as t increasesdepends on the initial value yo.b. Suppose that yo= 0.5. Find the time T at which the solutionfirst reaches the value 3.98.9Consider the initial value problemty(4- y)y' =1+ta. Determine how the solution behaves as t →∞.b. If yo = 2, find the time T at which the solution first reachesthe value 3.99.c. Find the range of initial values for which the solution lies inthe interval 3.99

Question

9 In each of Problems 1 through 8, solve the given differential equation.x²yy' + y² sinx = 0y' = cos²(x) cos² (2y)xy' = (1 - y²) 1/2dyx-e-xdxy + eyx²1+ y²1.2.3.4.5.6.7.y' =dydxdydxdydx==باد8.In each of Problems 9 through 16:form.a. Find the solution of the given initial value problem in explicitGb. Plot the graph of the solution.c. Determine (at least approximately) the interval in which thesolution is defined.9. y' = (1-2x) y²,10. y' = (1-2x)/y,11. xdx+ye-*dy = 0, y(0) = 112. dr/d0 = r²/0, r(1) = 213. y'=xy³ (1+x²)-1/2, y(0) = 114. y' = 2x/(1+2y), y(2) = 015. y' = (3x² - e*)/(2y-5), y(0) = 116. sin(2x) dx + cos(3y) dy = 0, y(π/2) = π/3Some of the results requested in Problems 17 through 22 can beobtained either by solving the given equations analytically or byplotting numerically generated approximations to the solutions. Tryto form an opinion about the advantages and disadvantages of eachapproach.G 17. Solve the initial value problemy'y(0) = -1/6y(1) = -2=y' ==1+3x²Зу2 — буand determine the interval in which the solution is valid.Hint: To find the interval of definition, look for points where theintegral curve has a vertical tangent.G 18. Solve the initial value problemy(0) = 13x²3y² - 4'y(1) = 0and determine the interval in which the solution is valid.Hint: To find the interval of definition, look for points where theintegral curve has a vertical tangent.G 19. Solve the initial value problemand determine where the solution attains its minimum value.G 20. Solve the initial value problem2- et3+2yy' = 2y² + xy², y(0) = 1y(0) = 0and determine where the solution attains its maximum value.G 21. Consider the initial value problemG 22.y' =y' =dQdt=ty(4- y)3a. Determine how the behavior of the solution as t increasesdepends on the initial value yo.b. Suppose that yo= 0.5. Find the time T at which the solutionfirst reaches the value 3.98.9Consider the initial value problemty(4- y)y' =1+ta. Determine how the solution behaves as t →∞.b. If yo = 2, find the time T at which the solution first reachesthe value 3.99.c. Find the range of initial values for which the solution lies inthe interval 3.99 < y < 4.01 by the time t = 2.23. Solve the equationloa taidanos ay(0) = yo.=&#34;, y(0) = yo > 0.dyay + bdx cy + dwhere a, b, c, and d are constants.24. Use separation of variables to solve the differential equation=r(a+bQ), Q(0) = 20,behaves as t→∞where a, b, r, and Qo are constants. Determine how the solutiondy/dxHomogeneous Equations. If the right-hand side of the equationf(x, y) can be expressed as a function of the ratio y/xonly, then the equation is said to be homogeneous.¹ Such equationscan always be transformed into separable equations by a change ofhomogeneous equations.the dependent variable. Problem 25 illustrates how to solve first-ordercontexts. The homogeneous equations considered here have nothing to do withThe word &#34;homogeneous&#34; has different meanings in different mathematicalthe homogeneous equations that will occur in Chapter 3 and elsewhere.

Accepted Answer

Given the equation and boundary condition. 
Required: Solution of equation and its graph.

In each of Problems 9 through 16: a. Find the solution of the given initial value problem in explicit form. G b. Plot the graph of the solution. c. Determine (at least approximately) the interval in which the solution is defined. 9. y'=(1-2x) y², y(0) = -1/6 10