with(DifferentialGeometry): with(ExteriorDifferentialSystems):

DGsetup([x, y], M1);

$\mathrm{frame\; name:\; M1}$

DGsetup([t], P1);

$\mathrm{frame\; name:\; P1}$

Omega := [(x^2 + y^2)*dx - 2*x*y *dy];

$\mathrm{\Ω}:=\left[\left(x^2\+y^2\right)\mathrm{dx}-2xy\mathrm{dy}\right]$

Sigma := IntegralManifold(Omega, P1);

$\mathrm{\Σ}:=\left[x\=0\,y\=0\right]\,\left[x\=\mathrm{I}\mathrm{\_C1}\,y\=\mathrm{\_C1}\right]\,\left[x\=-\mathrm{I}\mathrm{\_C1}\,y\=\mathrm{\_C1}\right]\,\left[x\=-\frac{1}{2}\mathrm{\_C1}\+\frac{1}{2}\sqrt{{\mathrm{\_C1}}^2\+4{\mathrm{F2}\left(t\right)}^2}\,y\=\mathrm{F2}\left(t\right)\right]\,\left[x\=-\frac{1}{2}\mathrm{\_C1}-\frac{1}{2}\sqrt{{\mathrm{\_C1}}^2\+4{\mathrm{F2}\left(t\right)}^2}\,y\=\mathrm{F2}\left(t\right)\right]$

Pullback(Sigma[3], Omega);

$\left[0\mathrm{dt}\right]$

IntegralManifold(Omega, P1, ansatz = [x = t]);

$\left[x\=t\,y\=-\sqrt{t^2\+t\mathrm{\_C1}}\right]\,\left[x\=t\,y\=\sqrt{t^2\+t\mathrm{\_C1}}\right]$

phi := Transformation(P1, M1, [x = f(t), y = g(t)]);

$\mathrm{\φ}:=\left[x\=f\left(t\right)\,y\=g\left(t\right)\right]$

IntegralManifold(Omega, phi);

$\left[x\=0\,y\=g\left(t\right)\right]\,\left[x\=\mathrm{\_C1}\,y\=0\right]\,\left[x\=f\left(t\right)\,y\=-\sqrt{{f\left(t\right)}^2\+f\left(t\right)\mathrm{\_C1}}\right]\,\left[x\=f\left(t\right)\,y\=\sqrt{{f\left(t\right)}^2\+f\left(t\right)\mathrm{\_C1}}\right]$

phi := Transformation(P1, M1, [x = t, y = g(t)]);

$\mathrm{\φ}:=\left[x\=t\,y\=g\left(t\right)\right]$

IntegralManifold(Omega, phi);

$\left[x\=t\,y\=-\sqrt{t^2\+t\mathrm{\_C1}}\right]\,\left[x\=t\,y\=\sqrt{t^2\+t\mathrm{\_C1}}\right]$

phi2 := Transformation(P1, M1, [x = r(t)*cos(theta(t)), y = r(t)*sin(theta(t))]);

$\mathrm{\φ2}:=\left[x\=r\left(t\right)\cos \left(\mathrm{\θ}\left(t\right)\right)\,y\=r\left(t\right)\sin \left(\mathrm{\θ}\left(t\right)\right)\right]$

Sigma := IntegralManifold(Omega, phi2);

$\mathrm{\Σ}:=$

phi3 := Transformation(P1, M1, [x = r(t)*cos(t), y = r(t)*sin(t)]);

$\mathrm{\φ3}:=\left[x\=r\left(t\right)\cos \left(t\right)\,y\=r\left(t\right)\sin \left(t\right)\right]$

IntegralManifold(Omega, phi3);

$\left[x\=0\,y\=0\right]\,\left[x\=\frac{\mathrm{\_C1}{\cos \left(t\right)}^2}{\cos \left(2t\right)}\,y\=\frac{\mathrm{\_C1}\cos \left(t\right)\sin \left(t\right)}{\cos \left(2t\right)}\right]$

IntegralManifold(Omega, phi3, auxiliaryequations = {r(0) = 1});

$\left[x\=\frac{{\cos \left(t\right)}^2}{\cos \left(2t\right)}\,y\=\frac{\cos \left(t\right)\sin \left(t\right)}{\cos \left(2t\right)}\right]$

DGsetup([x, y, z], M2);

$\mathrm{frame\; name:\; M2}$

DGsetup([t], P1);

$\mathrm{frame\; name:\; P1}$

DGsetup([s, t], P2);

$\mathrm{frame\; name:\; P2}$

Omega := [(y + z)*dx + (x + z)*dy + (x + y)*dz];

$\mathrm{\Ω}:=\left[\left(y\+z\right)\mathrm{dx}\+\left(x\+z\right)\mathrm{dy}\+\left(x\+y\right)\mathrm{dz}\right]$

phi1 := Transformation(P1, M2, [x = f(t), y = g(t), z = k(t)]);

$\mathrm{\φ1}:=\left[x\=f\left(t\right)\,y\=g\left(t\right)\,z\=k\left(t\right)\right]$

IntegralManifold(Omega, phi1);

$\left[x\=0\,y\=0\,z\=k\left(t\right)\right]\,\left[x\=\mathrm{\_C1}\,y\=-\mathrm{\_C1}\,z\=k\left(t\right)\right]\,\left[x\=f\left(t\right)\,y\=g\left(t\right)\,z\=\frac{-g\left(t\right)f\left(t\right)\+\mathrm{\_C1}}{f\left(t\right)\+g\left(t\right)}\right]$

phi2 := Transformation(P2, M2, [x = s, y = t, z = f(s, t)]);

$\mathrm{\φ2}:=\left[x\=s\,y\=t\,z\=f\left(s\,t\right)\right]$

Sigma := IntegralManifold(Omega, phi2);

$\mathrm{\Σ}:=\left[x\=s\,y\=t\,z\=\frac{-ts\+\mathrm{\_C1}}{s\+t}\right]$

Pullback(Sigma, Omega);

$\left[0\mathrm{ds}\right]$

DGsetup([x], P3);

$\mathrm{frame\; name:\; P3}$

DGsetup([x, y, y1, y2], M3);

$\mathrm{frame\; name:\; M3}$

Omega := evalDG([dy - y1*dx, dy1- y2*dx, dy2 - y1*dx]);

$\mathrm{\Ω}:=\left[-\mathrm{y1}\mathrm{dx}\+\mathrm{dy}\,-\mathrm{y2}\mathrm{dx}\+\mathrm{dy1}\,-\mathrm{y1}\mathrm{dx}\+\mathrm{dy2}\right]$

IntegralManifold(Omega, P3, ansatz = [x = x]);

$\left[x\=x\,y\=\mathrm{\_C1}{\ⅇ}^{-x}\+\mathrm{\_C2}{\ⅇ}^x\+\mathrm{\_C3}\,\mathrm{y1}\=-\mathrm{\_C1}{\ⅇ}^{-x}\+\mathrm{\_C2}{\ⅇ}^x\,\mathrm{y2}\=\mathrm{\_C1}{\ⅇ}^{-x}\+\mathrm{\_C2}{\ⅇ}^x\right]$

DGsetup([x, y], P4);

$\mathrm{frame\; name:\; P4}$

DGsetup([x, y, u, p, q], M4);

$\mathrm{frame\; name:\; M4}$

Omega := evalDG([du -p*dx -q*dy, (dp - 2*u/(x+y)^2*dy) &w dx , (dq - 2*u/(x+y)^2*dx) &w dy]);

$\mathrm{\Ω}:=\left[-p\mathrm{dx}-q\mathrm{dy}\+\mathrm{du}\,\frac{2u\mathrm{dx}\bigwedge \mathrm{dy}}{{\left(x\+y\right)}^2}-\mathrm{dx}\bigwedge \mathrm{dp}\,-\frac{2u\mathrm{dx}\bigwedge \mathrm{dy}}{{\left(x\+y\right)}^2}-\mathrm{dy}\bigwedge \mathrm{dq}\right]$

Sigma := IntegralManifold(Omega, P4, ansatz = [x = x, y = y]);

$\mathrm{\Σ}:=\left[x\=x\,y\=y\,u\=\frac{\left(x\+y\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{\_F1}\left(x\right)\right)\+\left(x\+y\right)\left(\frac{\ⅆ}{\ⅆy}\mathrm{\_F2}\left(y\right)\right)-2\mathrm{\_F2}\left(y\right)-2\mathrm{\_F1}\left(x\right)}{2x\+2y}\,p\=\frac{1}{2}\frac{{\left(x\+y\right)}^2\left(\frac{\ⅆ}{\ⅆx}\left(\frac{\ⅆ}{\ⅆx}\mathrm{\_F1}\left(x\right)\right)\right)\+\left(-2x-2y\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{\_F1}\left(x\right)\right)\+2\mathrm{\_F2}\left(y\right)\+2\mathrm{\_F1}\left(x\right)}{{\left(x\+y\right)}^2}\,q\=\frac{1}{2}\frac{{\left(x\+y\right)}^2\left(\frac{\ⅆ}{\ⅆy}\left(\frac{\ⅆ}{\ⅆy}\mathrm{\_F2}\left(y\right)\right)\right)\+\left(-2x-2y\right)\left(\frac{\ⅆ}{\ⅆy}\mathrm{\_F2}\left(y\right)\right)\+2\mathrm{\_F2}\left(y\right)\+2\mathrm{\_F1}\left(x\right)}{{\left(x\+y\right)}^2}\right]$

Pullback(Sigma, Omega);

$\left[0\mathrm{dx}\,0\mathrm{dx}\bigwedge \mathrm{dy}\,0\mathrm{dx}\bigwedge \mathrm{dy}\right]$