"File ODE1.MTH, copyright (c) 1991 by Soft Warehouse, Inc."

LINEAR1(p,q,x,y,x0,y0):=y=(y0+INT(q*#e^INT(p,x,x0,x),x,x0,x))/#e^INT(p,x,x0,x)

LINEAR1_GEN(p,q,x,y,c):=y=(c+INT(q*#e^INT(p,x),x))/#e^INT(p,x)

SEPARABLE(p,q,x,y,x0,y0):=INT(1/q,y,y0,y)=INT(p,x,x0,x)

SEPARABLE_GEN(p,q,x,y,c):=INT(1/q,y)=INT(p,x)+c

HOMOGENEOUS(r,x,y,x0,y0):=IF(DIF(LIM(r,y,x*y),x),LIM(SEPARABLE(1/x,LIM(r,x,1)-~
y,x,y,x0,y0/x0),y,y/x),"inapplicable","inapplicable")

HOMOGENEOUS_GEN(r,x,y,c):=IF(DIF(LIM(r,y,x*y),x),LIM(SEPARABLE_GEN(1/x,LIM(r,x~
,1)-y,x,y,c),y,y/x),"inapplicable","inapplicable")

EXACT(p,q,x,y,x0,y0):=IF(DIF(q,x)=DIF(p,y),INT(LIM(q,x,x0),y,y0,y)+INT(p,x,x0,~
x)=0,"inapplicable","inapplicable")

EXACT_GEN_AUX(s,q,y):=s+INT(q-DIF(s,y),y)

EXACT_GEN(p,q,x,y,c):=IF(DIF(q,x)=DIF(p,y),EXACT_GEN_AUX(INT(p,x),q,y)=c,~
"inapplicable","inapplicable")

BERNOULLI(p,q,k,x,y,x0,y0):=LINEAR1((1-k)*p,(1-k)*q,x,y^(1-k),x0,y0^(1-k))

BERNOULLI_GEN(p,q,k,x,y,c):=LINEAR1_GEN((1-k)*p,(1-k)*q,x,y^(1-k),c)

INTF_AUX(mu,p,q,x,y,x0,y0):=INT(LIM(mu*q,x,x0),y,y0,y)+INT(mu*p,x,x0,x)=0

INTF_HLP(p,q,x,y,x0,y0):=~
 IF(DIF((DIF(p,y)-DIF(q,x))/q,y),~
    INTF_AUX(#e^INT((DIF(p,y)-DIF(q,x))/q,x),p,q,x,y,x0,y0),~
    "inapplicable",~
    "inapplicable")

INTEGRATING_FACTOR(p,q,x,y,x0,y0):=~
 IF(DIF((DIF(q,x)-DIF(p,y))/p,x),~
    INTF_AUX(#e^INT((DIF(q,x)-DIF(p,y))/p,y),p,q,x,y,x0,y0),~
    INTF_HLP(p,q,x,y,x0,y0),~
    INTF_HLP(p,q,x,y,x0,y0))

INTF_GEN_AUX(mu,p,q,x,y):=EXACT_GEN_AUX(INT(mu*p,x),mu*q,y)

INTF_GEN_HLP(p,q,x,y,c):=~
 IF(DIF((DIF(p,y)-DIF(q,x))/q,y),~
    INTF_GEN_AUX(#e^INT((DIF(p,y)-DIF(q,x))/q,x),p,q,x,y)=c,~
    "inapplicable",~
    "inapplicable")

INTEGRATING_FACTOR_GEN(p,q,x,y,c):=~
 IF(DIF((DIF(q,x)-DIF(p,y))/p,x),~
    INTF_GEN_AUX(#e^INT((DIF(q,x)-DIF(p,y))/p,y),p,q,x,y)=c,~
    INTF_GEN_HLP(p,q,x,y,c),~
    INTF_GEN_HLP(p,q,x,y,c))

FUN_LIN_CCF(r,p,q,k,x,y,x0,y0):=LIM(SEPARABLE(1,p+q*LIM(r,y,(y-p*x-k)/q),x,y,x~
0,p*x0+q*y0+k),y,p*x+q*y+k)

FUN_LIN_CCF_GEN(r,p,q,k,x,y,c):=LIM(SEPARABLE_GEN(1,p+q*LIM(r,y,(y-p*x-k)/q),x~
,y,c),y,p*x+q*y+k)

LIN_FRAC_AUX(r,p,q,x,y,x0,y0):=LIM(HOMOGENEOUS(LIM(r,[y,x],[y+q,x+p]),x,y,x0-p~
,y0-q),[y,x],[y-q,x-p])

LIN_FRAC(r,a,b,d,p,q,k,x,y,x0,y0):=LIN_FRAC_AUX(r,(b*k-d*q)/(a*q-b*p),(d*p-a*k~
)/(a*q-b*p),x,y,x0,y0)

LIN_FRAC_AUX_GEN(r,p,q,x,y,c):=LIM(HOMOGENEOUS_GEN(LIM(r,[y,x],[y+q,x+p]),x,y,~
c),[y,x],[y-q,x-p])

LIN_FRAC_GEN(r,a,b,d,p,q,k,x,y,c):=LIN_FRAC_AUX_GEN(r,(b*k-d*q)/(a*q-b*p),(d*p~
-a*k)/(a*q-b*p),x,y,c)

GEN_HOM_AUX(k,r,x,y,x0,y0):=IF(DIF(k,x)=0 AND DIF(k,y)=0,LIM(SEPARABLE(1/x,y*(~
k+LIM(r*x/y,y,y/x^k)),x,y,x0,y0*x0^k),y,y*x^k),"inapplicable","inapplicable")

GEN_HOM(r,x,y,x0,y0):=GEN_HOM_AUX(x*DIF(x*r/y,x)/(y*DIF(x*r/y,y)),r,x,y,x0,y0)

GEN_HOM_AUX_GEN(k,r,x,y,c):=IF(DIF(k,x)=0 AND DIF(k,y)=0,LIM(SEPARABLE_GEN(1/x~
,y*(k+LIM(r*x/y,y,y/x^k)),x,y,c),y,y*x^k),"inapplicable","inapplicable")

GEN_HOM_GEN(r,x,y,c):=GEN_HOM_AUX_GEN(x*DIF(x*r/y,x)/(y*DIF(x*r/y,y)),r,x,y,c)

ALMOST_LIN(r,b,p,q,x,y,x0,y0):=IF(DIF(DIF(b,y)/r,y),LIM(LINEAR1(p*DIF(b,y)/r,q~
*DIF(b,y)/r,x,y,x0,LIM(b,y,y0)),y,b),"inapplicable","inapplicable")

ALMOST_LIN_GEN(r,b,p,q,x,y,c):=IF(DIF(DIF(b,y)/r,y),LIM(LINEAR1_GEN(p*DIF(b,y)~
/r,q*DIF(b,y)/r,x,y,c),y,b),"inapplicable","inapplicable")

TESTM2(r,x,y):=x^ELEMENT(r,1)*y^ELEMENT(r,2)

TESTM1(r,x,y):=TESTM2(ELEMENT(ROW_REDUCE([LIM(r,[y,x],[5,2]),LIM(r,[y,x],[3,7]~
)])`,3),x,y)

MONOMIAL_TEST(p,q,x,y):=TESTM1([y*q,-x*p,x*y*(DIF(p,y)-DIF(q,x))],x,y)

CLAIRAUT(p,q,x,y,v,c):=[LIM(p-q,v,c)=0,x*LIM(DIF(LIM(p,y,x*v-y),y),y,x*v-y)-~
DIF(q,v)=0]

DSOLVE1_GEN(p,q,x,y,c,a_):=~
 IF("inapplicable"=a_:=INTEGRATING_FACTOR_GEN(p,q,x,y,c),~
    IF("inapplicable"=a_:=HOMOGENEOUS_GEN(-p/q,x,y,c),~
       GEN_HOM_GEN(-p/q,x,y,c),a_,a_),a_,a_)

DSOLVE1(p,q,x,y,x0,y0,a_):=~
 IF("inapplicable"=a_:=INTEGRATING_FACTOR(p,q,x,y,x0,y0),~
    IF("inapplicable"=a_:=HOMOGENEOUS(-p/q,x,y,x0,y0),~
       GEN_HOM(-p/q,x,y,x0,y0),a_,a_),a_,a_)