Chapter 2
Solving First-Order Equations 2.1
Separable Equations
1. Use the method of separation of variables to solve each of these ordinary differential equations. dy (a) Write the equation x5 y 0 + y 5 = 0 in Leibnitz form x5 dx + y5 = 0 R R dx dy dy and separate the variables: y5 = − dx x5 . Integrate, y5 = − x5 , to obtain the solution: y −4 /(−4) = x−4 /4+C. This can also be written 4 in the form x4 + y 4 = Cx4 y 4 or y = ( Cxx4 −1 )1/4 . dy (b) Write the equation y 0 = 4xy in Leibnitz form dx = 4xy and separate R R dy dy the variables: y = 4xdx. Integrate, 4xdx, to obtain the y = solution: ln |y| = 2x2 + C. This can also be written in the form 2 y = Ce2x . dy + y tan x = 0 (c) Write the equation y 0 + y tan x = 0 in Leibnitz form dx R dy and separate the variables: y = − tan xdx. Integrate, dy y == R tan xdx, to obtain the solution: ln |y| = ln | cos x| + C. This can also be written in the form y = C cos x.
(d) The equation (1 + xR2 )dy + (1R+ y 2 )dx = 0 can be rearranged and dy dx integrated directly, 1+y 2 + 1+x2 = C. Therefore, the implicit solution is arctan y + arctan x = C. This can also be written in the form y = tan(C − arctan x). (e) Proceed as in part (d). Rearrange − xdy = 0 to the form R dx y lnR ydx dy dy dx − = 0 and integrate: − x y ln y x y ln y = C. This yields the implicit solution ln |x| − ln | ln y| = C which can also be written in the form ln y = Cx or y = eCx . 11
