Solve quetion 3.5.7.9 dx= x + y?dydx- x = ydyoris recognized as linear in the variable x. You should verify that the integratingfactor e-1)dy = e¯y and integration by parts yield the explicit solutionx = -y² – 2y - 2 + ce" for the second equation. This expression is, then,an implicit solution of the first equation.(iii) Mathematicians have adopted as their own certain words from engineer-ing, which they found appropriately descriptive. The word transient, usedearlier, is one of these terms. In future discussions the words input and outputwill occasionally pop up. The function f in (2) is called the input or drivingfunction; a solution y(x) of the differential equation for a given input is calledthe output or response.(iv) The term special functions mentioned in conjunction with the error func-tion also applies to the sine integral function and the Fresnel sine integralintroduced in Problems 49 and 50 in Exercises 2.3. "Special Functions" isactually a well-defined branch of mathematics. More special functions arestudied in Section 6.3.*Certain commands have the same spelling, but in Mathematica commands begin with a capital letter(Dsolve), whereas in Maple the same command begins with a lower case letter (dsolve). Whendiscussing such common syntax, we compromise and write, for example, dsolve. See the StudentResource and Solutions Manual for the complete input commands used to solve a linear first-order DE.EXERCISES 2.3Answers to selected odd-numbered problems begin on page ANS-2.In Problems 1-24 find the general solution of the given dif-ferential equation. Give the largest interval I over which thegeneral solution is defined. Determine whether there are anytransient terms in the general solution.5. y' + 3x²y = x²6. y' + 2xy = x³7. х?у + ху 3 18. y' = 2y + x² + 5dydy9. хy = x² sin x10. х+ 2y = 3dydydxdx1.= 5y2.+ 2y = 0dxdxdydy12. (1 + x) – xy = x + x²dx11. x+ 4y = x³ – xdydy´dx3.+ y = e3r4. 3+ 12y = 4dxdx13. x?y' + x(x + 2)y = e*2.3 LINEAR EQUATIONS61dy14. xy' + (1 + x)y = e¬* sin 2x33.+ 2xy = f(x), y(0) = 2, wheredx15. у dx — 4(х + yб) dy 3D 0(x, 0 1.dr21.+ r sec 0 = cos 0deUse a graphing utility to graph the continuous functiony(x).dP22.+ 2tP = P + 4t – 2dt36. Consider the initial-value problem y' + e*y = f(x),y(0) = 1. Express the solution of the IVP for x0 as anonelementary integral when f(x) = 1. What is the so-lution when f(x) = 0? When f(x) = e*?dy23. х+ (3x + 1)y = e¯3xdxdy24. (x² – 1)+ 2y = (x + 1)²dx37. Express the solution of the initial-value problemy' – 2xy = 1, y(1) = 1, in terms of erf(x).In Problems 25– 30 solve the given initial-value problem.Give the largest interval I over which the solution is defined.25. ху' + у %3D e', у(1) %3 2Discussion Problemsdx38. Reread the discussion following Example 2. Construct alinear first-order differential equation for which allnonconstant solutions approach the horizontal asymp-tote y = 4 as x→ 0.26. У- x =2y², y(1) = 5dydi+ Ri = E, i(0) = io,dt27. L39. Reread Example 3 and then discuss, with referenceto Theorem 1.2.1, the existence and uniqueness of asolution of the initial-value problem consisting ofxy' – 4y = x°e* and the given initial condition.(b) у(0) 3D yо, yo 0L, R, E, and io constantsLP= k(T – T„); T(0) = T0,dt28.k, Tm, and To constants(а) у(0) — 0dy(c) y(xo) = yo, xo 0, yo 0+ y = In x, y(1) = 10dx29. (х + 1)40. Reread Example 4 and then find the general solution ofthe differential equation on the interval (–3, 3).30. y' + (tan x)y = cos²x, y(0) = -1

Question

Solve quetion 3.5.7.9 dx= x + y?dydx- x = ydyoris recognized as linear in the variable x. You should verify that the integratingfactor e-1)dy = e¯y and integration by parts yield the explicit solutionx = -y² – 2y - 2 + ce&#34; for the second equation. This expression is, then,an implicit solution of the first equation.(iii) Mathematicians have adopted as their own certain words from engineer-ing, which they found appropriately descriptive. The word transient, usedearlier, is one of these terms. In future discussions the words input and outputwill occasionally pop up. The function f in (2) is called the input or drivingfunction; a solution y(x) of the differential equation for a given input is calledthe output or response.(iv) The term special functions mentioned in conjunction with the error func-tion also applies to the sine integral function and the Fresnel sine integralintroduced in Problems 49 and 50 in Exercises 2.3. &#34;Special Functions&#34; isactually a well-defined branch of mathematics. More special functions arestudied in Section 6.3.*Certain commands have the same spelling, but in Mathematica commands begin with a capital letter(Dsolve), whereas in Maple the same command begins with a lower case letter (dsolve). Whendiscussing such common syntax, we compromise and write, for example, dsolve. See the StudentResource and Solutions Manual for the complete input commands used to solve a linear first-order DE.EXERCISES 2.3Answers to selected odd-numbered problems begin on page ANS-2.In Problems 1-24 find the general solution of the given dif-ferential equation. Give the largest interval I over which thegeneral solution is defined. Determine whether there are anytransient terms in the general solution.5. y' + 3x²y = x²6. y' + 2xy = x³7. х?у + ху 3 18. y' = 2y + x² + 5dydy9. хy = x² sin x10. х+ 2y = 3dydydxdx1.= 5y2.+ 2y = 0dxdxdydy12. (1 + x) – xy = x + x²dx11. x+ 4y = x³ – xdydy´dx3.+ y = e3r4. 3+ 12y = 4dxdx13. x?y' + x(x + 2)y = e*2.3 LINEAR EQUATIONS61dy14. xy' + (1 + x)y = e¬* sin 2x33.+ 2xy = f(x), y(0) = 2, wheredx15. у dx — 4(х + yб) dy 3D 0(x, 0 1.dr21.+ r sec 0 = cos 0deUse a graphing utility to graph the continuous functiony(x).dP22.+ 2tP = P + 4t – 2dt36. Consider the initial-value problem y' + e*y = f(x),y(0) = 1. Express the solution of the IVP for x>0 as anonelementary integral when f(x) = 1. What is the so-lution when f(x) = 0? When f(x) = e*?dy23. х+ (3x + 1)y = e¯3xdxdy24. (x² – 1)+ 2y = (x + 1)²dx37. Express the solution of the initial-value problemy' – 2xy = 1, y(1) = 1, in terms of erf(x).In Problems 25– 30 solve the given initial-value problem.Give the largest interval I over which the solution is defined.25. ху' + у %3D e', у(1) %3 2Discussion Problemsdx38. Reread the discussion following Example 2. Construct alinear first-order differential equation for which allnonconstant solutions approach the horizontal asymp-tote y = 4 as x→ 0.26. У- x =2y², y(1) = 5dydi+ Ri = E, i(0) = io,dt27. L39. Reread Example 3 and then discuss, with referenceto Theorem 1.2.1, the existence and uniqueness of asolution of the initial-value problem consisting ofxy' – 4y = x°e* and the given initial condition.(b) у(0) 3D yо, yo > 0L, R, E, and io constantsLP= k(T – T„); T(0) = T0,dt28.k, Tm, and To constants(а) у(0) — 0dy(c) y(xo) = yo, xo > 0, yo > 0+ y = In x, y(1) = 10dx29. (х + 1)40. Reread Example 4 and then find the general solution ofthe differential equation on the interval (–3, 3).30. y' + (tan x)y = cos²x, y(0) = -1

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