Identify the following Differential Equations as homogenous, Bernoulli or of for
ID: 2985798 • Letter: I
Question
Identify the following Differential Equations as homogenous, Bernoulli or of form y' = G(ax+by), then solve:
a) dy/dx = ((x^2) + 3(y^2)) / (2xy)
b) dy/dx = (((x + y + 1)^2) - ((x + y - 1)^2))
Explanation / Answer
a) dy/dx = ((x^2) + 3(y^2)) / (2xy) it is homogeneous. dy/dx = ((x^2) + 3(y^2)) / (2xy) let y = vx dy/dx = v+ x dv/dx so, v + xdv/dx = [x^2(1 + 3v^2)]/[2x^2v] ==> v + xdv/dx = [(1 + 3v^2)]/[2v] ==> xdv/dx = [ 1 + v^2]/[2v] ==> dv * [2v]/[1+v^2] = dx /x integrate we get ln|(1+v^2)| = ln|x| + ln|C| ==> 1 + v^2 = Cx ==> x^2 + y^2 = Cx^3 b) dy/dx = (((x + y + 1)^2) - ((x + y - 1)^2)) it is of y' = G(ax+by) form let z = x + y dz/dx = 1 + dy/dx dy/dx = (((x + y + 1)^2) - ((x + y - 1)^2)) we get dz/dx - 1 = (((z + 1)^2) - ((z - 1)^2)) ==> dz/dx - 1 = 4z ==> dz/dx - 4z = 1 integrating factor = e^(-4x) solution is z * e^(-4x) = integral [ e^(-4x) * 1] dx + C ==> (x+y)*e^(-4x) = -e^(-4x) /4 + C ==> y = -x - 1/4 + Ce^(4x)