2.108   ODE No. 108

  1. Problem in Latex
  2. Mathematica input
  3. Maple input

xy(x)+y(x)+y(x)2(log(x))=0 Mathematica : cpu = 0.0105073 (sec), leaf count = 15

{{y(x)1c1x+log(x)+1}}

Maple : cpu = 0.011 (sec), leaf count = 13

{y(x)=(1+_C1x+ln(x))1}

Hand solution

xy+axy2+2y+bx=0This is Riccati non-linear first order. Converting it to standard form(1)xyy2lnx+y=0y=1xy+y2lnxx=f0+f1y+f2y2

This is Bernoulli non-linear first order ODE since f0=0. Dividing by y2 gives

yy2=1x1y+lnxx

Let u=1y, hence u=yy2, and the above becomes

u=1xu+lnxxu1xu=lnxx

Integrating factor is μ=e1xdx=elnx=1x, hence

d(μu)=μlnxx

Integrating

1xu=1x2lnxdx+C=(lnxx1x)+C

Therefore

u=lnx+1+Cx

Since u=1y then

y=1lnx+1+Cx

Verification

restart; 
ode:=x*diff(y(x),x)-y(x)^2*ln(x)+y(x)=0; 
my_solution:=1/(ln(x)+1+_C1*x); 
odetest(y(x)=my_solution,ode); 
0