Solve the general solution of the following.

Question

a. dy/dx = 1/2((x+y-1)/(x+2))²

b. xdy/dx = 6xe^2x+y(2x-1)

c. xy'+y²lnx+y = 0

Answer 1

a. dy/dx = 1/2((x+y-1)/(x+2))²

y' = 1/2((x+y-1)/(x+2))²

let u = (x+y-1)/(x+2)

solving for y: y = u(x+2)-x+1

plug in for y: (u(x+2)-x+1)' = (1/2)u²

u+u'(x+2)-1 = (1/2)u²

u'(x+2) = (1/2)u²-u+1

du/dx 2(x+2) = u²-2u+2

1/(u²-2u+2) du = 1/(2(x+2)) dx

1/((u-1)²+1) du = 1/(2(x+2)) dx

integrate^^^^^^^^^^^^^^^^^^^^

arctan(u-1) = (1/2)ln(x+2) + C

u = tan((1/2)ln(x+2) + C) + 1

substitute for u: (x+y-1)/(x+2) = tan((1/2)ln(x+2) + C) + 1

solve for y: x+y-1 = (x+2)(tan((1/2)ln(x+2) + C) + 1)

x+y-1 = xtan((1/2)ln(x+2) + C) + 2tan((1/2)ln(x+2) + C) + x + 2

y = xtan((1/2)ln(x+2) + C) + 2tan((1/2)ln(x+2) + C) + 3

y = (x+2)(tan((1/2)ln(x+2) + C)) + 3

b. xdy/dx = 6xe^2x+y(2x-1)

dy/dx = 6e^2x+y(2x-1)/x

dy/dx - y(2x-1)/x = 6e^2x

integrating factor: I = e^{integral(-(2x-1)/x)} = xe^-2x

dy/dx(xe^-2x) - (xe^-2x)y(2x-1)/x = 6e^2x(xe^-2x)

d/dx (yxe^-2x) = 6x

integrate ^^^^^^^

yxe^-2x = 3x² + C

y = 3xe^2x + Ce^2x/x

c. xy'+y²lnx+y = 0

rewrite in the form of bernoulli's ode: y' + yp(x) = yⁿq(x)

y'+y/x = -y²lnx/x

the power of y on the right hand side is n in bernoulli's equation

substitute: v = y^1-n and solve 1/(1-n)v' + v(px) = q(x)

v = y^-1

-v'+v(1/x) = -lnx/x

v'-v(1/x) = lnx/x

integrating factor: I = e^{integral(-1/x)} = 1/x

v'(1/x)-v(1/x)² = lnx/x²

d/dx (v/x) = lnx/x²

integrate^^^^^^^^

v/x = -lnx/x-1/x+C₁ <---u substitution

v = -lnx-1+C₁x

v = -(lnx+1+Cx)

substitute: v=y^-1

y^-1 = -(lnx+1+Cx)

y = -1/(lnx+1+Cx)