Dr. Z’s Math152 Handout #9.3 [Separable Equations]By Doron ZeilbergerProblem Type 9.3a: Solve the differential equationy0=A(x)B(y)or y0= A(x)B(y) etc.Example Problem 9.3a: Solve the differential equationy0= y2sec xSteps Example1. If not already written like this, replacey0bydydx. Treat dy and dx as algebraicquantities and separate the x part fromthe y part.dydx=A(x)B(y)ordydx= A(x)B(y) etc.impliesB(y)dy = A(x)dx respectivelydyB(y)= A(x)dx etc.1.dydx= y2sec ximpliesdyy2= sec x dxwhich is the same asy−2dy = sec x dx2. Apply the Integral sign to both sides,and perform the integration. Only putthe +C on one side.2.Zy−2dy =Zsec x dx ,gives−1y= ln | sec x + tan x| + C13. If possible, solve for y. Otherwise leaveit in implicit form. If there is an initialcondition then plug it in and solve for C.If nothing is mentioned (like in this prob-lem), then leave C alone.3.y =−1ln | sec x + tan x| + CAns.: y =−1ln | sec x+tan x|+C.Problem Type 9.3b: Find an eqation of the curve that passes through the point (a, b) and whoseslope 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) andwhose slope at (x, y) is y2/x3.Steps Example1. Slope is derivative, so set it equal todydx. Treat dy and dx as algebraic quanti-ties and separate the x part from the ypart.dydx=A(y)B(x)impliesdyA(y)=dxB(x)1.dydx=y2x3impliesdyy2=dxx3which is the same asy−2dy = x−3dx2. Apply the Integral sign to both sides,and perform the integration. Only putthe +C on one side.2.y−1−1=x−2−2+ Cwhich gives−1y=−12x2+ C23. Plug in the point (x = a, y = b) andsolve for C. Plug back that value for Cand try to express y in terms of x if pos-sible. Otherwise leave it in implicit form.3.−11=−12 · 12+ Cgiving C = −1/2. Incorporating that Cgives−1y=−12x2−12and algebra givesy =2x21 +