1.5 Problems Find general solutions of the differential equations in Prob- lems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. 1. y' + y = 2, y(0) = 0 3. y'+3y=2xe-3x 5. xy +2y=3x, y(1) = 5 6. xy' +5y=7x2, y(2) = 5 7. 2xy +y=10x 9. xy' y=x, y(1) = 7 11. xy + y = 3xy, y(1) = 0 12. xy'+3y=2x5, y(2) = 1 13. y'ye y(0) = 1 14. xy-3y=x3, y(1) = 10 15. y+2xy=x, y(0) = -2 16. y = (1-y) cos x, y(л) = 2 2. y'-2y=3e2x, y(0) = 0 4. y' -2xy = ex2 8. 3xy+y=12x 10. 2xy'-3y=9x3 17. (1+x)y + y = cos x, y(0) = 1 18. xy2y+x³ cos x 19. y' y cot x = cos x 20. y1+x+y+xy, y(0) = 0 21. xy' 3y+x4 cos x, y(2л) = 0 22. y=2xy + 3x² exp(x²), y(0) = 5 23. xy + (2x-3)y = 4x4 24. (x2+4)y' + 3xy = x, y(0) = 1 , y(0) = 1 25. (x²+1)+3x3y = 6x exp (-2x2), y(0 dx Solve the differential equations in Problems 26 through 28 by regarding y as the independent variable rather than x. 26. (1 – 4xy2)dx = dy =y3 27. (x+yey) = 1 dx dy = 1+ y² 28. (1 + 2xy) 29. Express the general solution of dy/dx = 1 + 2xy in terms of the error function 2 erf(x)=e² dt. 30. Express the solution of the initial value problem dy 2x √x = y + 2x cos x, y(1)=0 as an integral as in Example 3 of this section. Problems 31 and 32 illustrate-for the special order linear equations-techniques that will when we study higher-order linear equations in 31. (a) Show that ye(x) = Ce-P(x) dx is a general solution of dy/dx + P(x)y = that yp(x) = e− / P(x)dx [ (Q(x)e/ P is a particular solution of dy/dx + P (c) Suppose that ye(x) is any general soluti P(x)y = 0 and that yp(x) is any particu dy/dx+ P(x)y = Q(x). Show that y(x)= is a general solution of dy/dx + P(x)y = 32. (a) Find constants A and B such that yp( B cos x is a solution of dy/dx + y = 2 sin result of part (a) and the method of Probler general solution of dy/dx + y = 2 sin.x. initial value problem dy/dx + y = 2 sin.x. 33. A tank contains 1000 liters (L) of a solutio 100 kg of salt dissolved in water. Pure w into the tank at the rate of 5 L/s, and the uniform by stirring- is pumped out at the long will it be until only 10 kg of salt rema 34. Consider a reservoir with a volume of feet (ft³) and an initial pollutant concentra There is a daily inflow of 500 million ft³ pollutant concentration of 0.05% and an o flow of the well-mixed water in the reser will it take to reduce the pollutant conce reservoir to 0.10%? 35. Rework Example 4 for the case of Lake empties into the St. Lawrence River and from Lake Erie (via the Niagara River). ences are that this lake has a volume of 10 inflow-outflow rate of 410 km³/year.