Rutgers University MTH 152 - Math 152 Handout 9 Separable Equations (3 pages)

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Dr Z s Math152 Handout 9 3 Separable Equations By Doron Zeilberger Problem Type 9 3a Solve the differential equation y0 A x B y or y 0 A x B y etc Example Problem 9 3a Solve the differential equation y 0 y 2 sec x Steps Example 1 If not already written like this replace dy y 0 by dx Treat dy and dx as algebraic quantities and separate the x part from the y part 1 dy A x dx B y or dy A x B y etc dx dy y 2 sec x dx implies dy sec x dx y2 which is the same as y 2 dy sec x dx implies B y dy A x dx respectively dy A x dx B y etc 2 Apply the Integral sign to both sides and perform the integration Only put the C on one side 2 Z y 2 dy Z sec x dx gives 1 ln sec x tan x C y 1 3 If possible solve for y Otherwise leave it in implicit form If there is an initial condition then plug it in and solve for C If nothing is mentioned like in this problem then leave C alone 3 y Ans y 1 ln sec x tan x C 1 ln sec x tan x C Problem Type 9 3b Find an eqation of the curve that passes through the point a b and whose slope at x y is A y B x Example Problem 9 3b Find an eqation of the curve that passes through the point 1 1 and whose slope at x y is y 2 x3 Steps Example 1 Slope is derivative so set it equal to dy dx Treat dy and dx as algebraic quantities and separate the x part from the y part dy A y dx B x 1 dy y2 3 dx x implies dy dx 3 y2 x which is the same as implies dx dy A y B x 2 Apply the Integral sign to both sides and perform the integration Only put the C on one side y 2 dy x 3 dx 2 x 2 y 1 C 1 2 which gives 1 1 2 C y 2x 2 3 Plug in the point x a y b and solve for C Plug back that value for C and try to express y in terms of x if possible Otherwise leave it in implicit form 3 1 1 C 1 2 12 giving C 1 2 Incorporating that C gives 1 1 1 2 y 2x 2 and algebra gives y Ans 3 2x2 1 x2 2x2 1 x2