dy/dx+yx^-1=tan^-1(x). Let p(x)=x^-1 and q(x)=tan^-1(x).
Multiply through by an unknown function u(x):
u(x)dy/dx+u(x)yp(x)=u(x)q(x).
d/dx(uy)=udy/dx+ydu/dx or u(x)dy/dx+ydu/dx.
Comparing this expansion of the differentiation of a product, we have du/dx=u(x)p(x)=u(x)/x.
So, du/u=dx/x; integrating: ln(u)=ln(x)+k, where k is a constant which can be written k=ln(a) and u(x)=ax, where a is a constant.
Now we have d/dx(axy)=axdy/dx+ay=a(xdy/dx+y)=ax(dy/dx+y/x)=axtan^-1(x).
Integrating wrt x axy=integral(axtan^-1(x)dx), xy=integral(xtan^-1(x)dx).
A slight diversion:
(1) integral(tan^-1(x)dx). Let r=tan^-1(x), x=tan(r), dx=sec^2(r)dr;
integral(tan^-1(x)dx)=integral(rsec^2(r)dr);
let U=r, dU=dr, let dV=sec^2(r)dr, V=tan(r);
d/dr(UV)=UdV/dr+VdU/dr=rsec^2(r)+tan(r); UV=rtan(r)=integral(rsec^2(r)dr)+integral(sin(r)/cos(r).dr).
Therefore, integral(rsec^2(r)dr)=rtan(r)-integral(sin(r)/cos(r).dr)=rtan(r)+ln(cos(r)).
tan(r)=x so cos(r)=1/sqrt(1+x^2). Substituting for r:
integral(tan^-1(x)dx)=xtan^-1(x)+ln(1/sqrt(1+x^2))=xtan^-1(x)-(1/2)ln(1+x^2).
(2) integral(ln(1+x^2)dx). Let U=ln(1+x^2), dU=2x/(1+x^2); let dV=dx, V=x.
d/dx(UV)=UdV/dx+VdU/dx; UV=xln(1+x^2)=integral(ln(1+x^2)dx)+integral(2x^2/(1+x^2).dx)=
integral(ln(1+x^2)dx)+integral(((2x^2+2)/(1+x^2)-2/(1+x^2))dx)=
integral(ln(1+x^2)dx)+integral(2dx)-integral(2/(1+x^2).dx)=
integral(ln(1+x^2)dx)+2x-2tan^-1(x).
Therefore, integral(ln(1+x^2)dx)=xln(1+x^2)-2x+2tan^-1(x).
Back to the problem:
integral(xtan^-1(x)dx): let u=x, du=dx; dv=tan^-1(x)dx, v=xtan^-1(x)-(1/2)ln(1+x^2).
d/dx(uv)=udv/dx+vdu/dx;
uv=x(xtan^-1(x)-(1/2)ln(1+x^2))=
integral(xtan^-1(x)dx)+integral((xtan^-1(x)-(1/2)ln(1+x^2))dx)=
2*integral(xtan^-1(x)dx)-(1/2)integral(ln(1+x^2)dx)=
2*integral(xtan^-1(x)dx)-(1/2)(xln(1+x^2)-2x+2tan^-1(x)).
Therefore:
2*integral(xtan^-1(x)dx)=x(xtan^-1(x)-(1/2)ln(1+x^2))+(1/2)(xln(1+x^2)-2x+2tan^-1(x)).
2*integral(xtan^-1(x)dx)=x(xtan^-1(x)-(1/2)ln(1+x^2))+(1/2)(xln(1+x^2)-2x+2tan^-1(x))=
x^2tan^-1(x)-x+tan^-1(x)=(tan^-1(x))(1+x^2)-x.
Therefore, integral(xtan^-1(x)dx)=(1/2)((1+x^2)tan^-1(x)-x) and
xy=(1/2)((1+x^2)tan^-1(x)-x) and y=(1/2x)((1+x^2)tan^-1(x)-x).