By Rudolph E. Langer

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**Additional info for A First Course in Ordinary Differential Equations**

**Example text**

A 10-lb. body whose specific heat is -g- is at 0 degrees when t ^ 0. It is then plunged into 100 lb- of water at 80 degrees. 6 applies with k — yf* what is the temperature of the body at time t? 43. A 5-lb. body whose specific heat is TnJ" is plunged at / 0 into 50 lb. of water. If the temperature of the body at this time is 200 degrees^ and if it eventually cools to 50 degrees, what is the formula for its temperature at time /? 44. A 50-lb. body whose specific heat is i^f, and whose temperature is 208 dc^ecs, is plunged into 20 lb.

22) dx/dt = Q{x,y), dy/dt = -P(x,y). T h e two equations that make up the general integral of this system express X and y i n terms o( t. T h e y therefore give a parametric representation of 30 . Some Types of Solvable Equations the integral of the original differential equation. T h a t integral i n terms of X and y is obtainable by eliminating / between the pair of equations which constitutes the integral of the system. 1 by the system dx/di = k{x,y)Q{x,y), dy/di = -k{x, y)P{x, y), with any chosen function k{x,y).

2x +y\ 32. {2xy2 -y^\dx-\r 33. \x^ 34. (4*2 ~ 35. [2x^y + > M - W xy^'dy = dy = 0. 0. xy -h 7x -\- 2y - X dx + sin y 36. 1 37. - sec y — tan y ^ ) dx X 38. - + {2x^ + I] dx + xy] dy = X log X {x - 6y + + ~e 0. 0. x^ COS y dy = 0. dx — {x — sec y log x] dy ~ dx \ ] dy " + JT sin ^ X 0. 5. 10) y' +p(x)y = q{x). It is called the linear differential equation of the first crder because it is linear, that is, of the first degree, i n y and y'. For an equation of this type an integrating factor is always easily found.