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z: s*x^2;
equation: z*(1-z)*Ypp + (c-(a+b+1)*z)*Yp-a*b*Y;
gradef(w(x), wp(x));
gradef(wp(x), wpp(x));
g: w(z);
gp: diff(g, x);
gpp: diff(gp, x);
Ag: matrix(
[ coeff(g, w(z)), coeff(g, wp(z)), coeff(g, wpp(z)) ],
[ coeff(gp, w(z)), coeff(gp, wp(z)), coeff(gp, wpp(z)) ],
[ coeff(gpp,w(z)), coeff(gpp,wp(z)), coeff(gpp,wpp(z)) ]
);
Agi: invert(Ag);
W: Agi.matrix([G],[Gp],[Gpp]);
eg: subst(W[1,1], Y, equation);
eg: subst(W[2,1], Yp, eg);
eg: subst(W[3,1], Ypp, eg);
eg: 4*s*x*eg;
eg: expand(ratsimp(eg));
ratsimp(subst(0, G, subst(0, Gp, eg)) / Gpp);
ratsimp(subst(0, G, subst(0, Gpp, eg)) / Gp);
ratsimp(subst(0, Gp, subst(0, Gpp, eg)) / G);
u: x*w(z);
up: diff(u, x);
upp: diff(up, x);
Au: matrix(
[ coeff(u, w(z)), coeff(u, wp(z)), coeff(u, wpp(z)) ],
[ coeff(up, w(z)), coeff(up, wp(z)), coeff(up, wpp(z)) ],
[ coeff(upp,w(z)), coeff(upp,wp(z)), coeff(upp,wpp(z)) ]
);
Aui: invert(Au);
W: Aui.matrix([U],[Up],[Upp]);
eu: subst(W[1,1], Y, equation);
eu: subst(W[2,1], Yp, eu);
eu: subst(W[3,1], Ypp, eu);
eu: 4*s*x^3*eu;
eu: expand(ratsimp(eu));
display2d: false$
ratsimp(subst(0, U, subst(0, Up, eu)) / Upp);
ratsimp(subst(0, U, subst(0, Upp, eu)) / Up);
ratsimp(subst(0, Up, subst(0, Upp, eu)) / U);
display2d: true$
/* allgemeiner Fall, f(x) = x^nu w(s * x^rho) */
z: s*x^nu;
f: x^rho * w(z);
fp: diff(f, x);
fpp: diff(fp, x);
Af: ratsimp(matrix(
[ coeff(f, w(z)), coeff(f, wp(z)), coeff(f, wpp(z)) ],
[ coeff(fp, w(z)), coeff(fp, wp(z)), coeff(fp, wpp(z)) ],
[ coeff(fpp,w(z)), coeff(fpp,wp(z)), coeff(fpp,wpp(z)) ]
));
Afi: invert(Af);
Afi: ratsimp(Afi);
W: Afi.matrix([F],[Fp],[Fpp]);
ef: subst(W[1,1], Y, equation);
ef: subst(W[2,1], Yp, ef);
ef: subst(W[3,1], Ypp, ef);
ef: s * nu^2 * ef;
ef: expand(ratsimp(ef));
display2d: true$
c2: ratsimp(subst(0, F, subst(0, Fp, ef)) / Fpp);
c1: ratsimp(subst(0, F, subst(0, Fpp, ef)) / Fp);
c0: ratsimp(subst(0, Fp, subst(0, Fpp, ef)) / F);
ratsimp(subst(1, s, subst(3/2, c, subst(1/2, b, subst(1/2, a,
subst(2, nu, subst(1, rho, c2)))))));
ratsimp(subst(1, s, subst(3/2, c, subst(1/2, b, subst(1/2, a,
subst(2, nu, subst(1, rho, c1)))))));
ratsimp(subst(1, s, subst(3/2, c, subst(1/2, b, subst(1/2, a,
subst(2, nu, subst(1, rho, c0)))))));
/* Differentialgleichung von 0F1 */
display2d: true$
equation0f1: z*Ypp + beta*Yp - Y;
e0f1: subst(W[1,1], Y, equation0f1);
e0f1: subst(W[2,1], Yp, e0f1);
e0f1: subst(W[3,1], Ypp, e0f1);
e0f1: s*nu^2*x^(rho+nu) * e0f1;
e0f1: expand(ratsimp(e0f1));
display2d: false$
ratsimp(subst(0, F, subst(0, Fp, e0f1)) / Fpp);
ratsimp(subst(0, F, subst(0, Fpp, e0f1)) / Fp);
ratsimp(subst(0, Fp, subst(0, Fpp, e0f1)) / F);
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