#10; psolution1(a_,n_,b_,m_,c_,d_,y_,t_,xx_,p_):= if(t==xx,if(isconstant(d),if(c==0,-d*t^p/p!, -d/c), if(hasnot(d,x) and has(d,t),psolution(-c,-d,y,t,p),0 )), if(hasnot(d,t) and has(d,xx),psolution(-b/a,m,-c/a,-d/a,y,xx,n),0 )); psolution(a_,n_,b_,m_,c_,c1_*c2_,y_,t_,x_,p_):=if(isconstant(c1),c1*psolution(a,n,b,m,c,c2,y,t,x,p),if(isconstant(c2),c2*psolution(a,n,b,m,c,c1,y,t,x,p) )); #psolution(a_,n_,b_,m_,c_,t_*ax_,y_,t_,x_,p_):= if(p>1, psolution(b/a,m,c/a,ax/a,y,x,n)*t, if(c==0,psolution(a,n-1,b,t*ax,y,t,p) )); #psolution(a_,n_,b_,m_,c_,tt_*x_,y_,t_,x_,p_):= if(p>1, psolution(-b/a,m,-c/a,-x/a,y,x,n)*t, psolution(-c,-tt,y,t,p)*x+if(m==1,-c*b*int(tt,t,p)/2,0) ); #psolution(a_,n_,b_,m_,c_,tt_*x_,y_,t_,x_,p_):= if(m>=1, psolution(-b,m,c,tt*x,y,t,x,p) ); #psolution(a_,n_,b_,m_,c_,t_*x_,y_,t_,x_,p_):= if(p>1, psolution(b/a,m,c/a,x/a,y,x,n)*t, psolution(-b,m,-c,-t*x,y,t,x,p) ); psolution(a_,n_,b_,m_,c_,t_*x_,y_,t_,x_,p_):= if(n>1, if(p>1, psolution(-c/b,-x/b,y,x,m)*t, psolution(b,m,-c,-t*x,y,t,x,p) )); psolution(a_,n_,b_,m_,c_,exp(x_)*tt_,y_,t_,x_,p_):= if(a-b-c==0 or isconstant(a,b,c), -psolution(-a-b-c,tt,y,t,p)*exp(x) ); #psolution(a_,n_,b_,m_,c_,exp(x_)*at_,y_,t_,x_,p_):= if(a-b-c==0 or isconstant(a,b,c), -psolution(-a-b-c,at,y,t,p)*exp(x) ); psolution(a_,n_,b_,m_,c_,exp(t_)*xx_,y_,t_,x_,p_):= if(isconstant(a,b,c), psolution(-b/a,m,(-1-c)/a,-xx/a,y,x,n)*exp(t) ); psolution(a_,n_,b_,m_,c_,exp(t_)*ax_,y_,t_,x_,p_):= if(isconstant(a,b,c), psolution(-b/a,m,(-1-c)/a,-ax/a,y,x,n)*exp(t) ); #psolution(a_,n_,b_,m_,c_,exp(t)*x_,y_,t_,x_,p_):= if(isconstant(a,b,c), if(c==0 and m==1, (b+x)*exp(t), psolution(c-1,x,y,x,m)*exp(t) )); #psolution(a_,n_,b_,m_,c_,exp(t)*x_,y_,t_,x_,p_):= if(isconstant(a,b,c), if(c==0 and m==1, (b+x)*exp(t), if(c==1, psolution(b,m,0,x,y,x,n)*exp(t), if(m>=1, psolution(-b,m,c,exp(t)*x,y,t,p) )))); psolution(a_,n_,b_,m_,c_,exp(t+x),y_,t_,x_,p_):= if(isconstant(b,c) and t<>x, -exp(t+x)/(1+a+b+c),0 ); psolution(a_,n_,b_,m_,c_,exp(t-x),y_,t_,x_,p_):= if(isconstant(b,c) and t<>x, -exp(t-x)/(1+a-b+c),0 ); psolution(a_,n_,b_,m_,c_,c1_*c2_*c3_,y_,t_,x_,p_):= if(isfree(c1,x) and hasnot(c1,t), c1*psolution(a,n,b,m,c,c2*c3,y,t,x,p), if(isfree(c3,x) and hasnot(c3,t), c3*psolution(a,n,b,m,c,c1*c2,y,t,x,p) )); #psolution(a_,n_,b_,m_,c_,c1_+c2_,y_,t_,x_,p_):= if(has(c1,t) and hasnot(c1,x), if(has(c2,x) and hasnot(c2,t), psolution(-b/a,m,-c/a,-c2/a,y,x,n),psolution(a,n,b,m,c,c2,y,t,x,p))+psolution(-c,-c1,y,t,p), if(has(c2,t) and hasnot(c2,x), if(has(c1,x) and hasnot(c1,t), psolution(-b/a,m,-c/a,-c1/a,y,x,n),psolution(a,n,b,m,c,c1,y,t,x,p))+psolution(-c,-c2,y,t,p), if(has(c1,x) and hasnot(c1,t), psolution(a,n,b,m,c,c2,y,t,x,p)+psolution(-b/a,m,-c/a,-c1/a,y,x,n), if(has(c2,x) and hasnot(c2,t), psolution(a,n,b,m,c,c1,y,t,x,p)+psolution(-b/a,m,-c/a,-c2/a,y,x,n), psolution(a,n,b,m,c,c1,y,t,x,p)+psolution(a,n,b,m,c,c2,y,t,x,p) )))); psolution(a_,n_,b_,m_,c_,c1_+c2_,y_,t_,x_,p_):= psolution(a,n,b,m,c,c1,y,t,x,p)+psolution(a,n,b,m,c,c2,y,t,x,p); #8; psolution1(b_,m_,c_,d_,y_,t_,xx_,p_):= if(t==xx,if(isconstant(d),if(c==0,-d*t^p/p!, -d/c), if(has(d,t) and hasnot(d,x),psolution(-c,-d,y,t,p),0 )), if(hasnot(d,t) and has(d,xx),psolution(-c/b,-d/b,y,xx,m), 0 )); #psolution1(b_,m_,c_,d_,y_,t_,x_,p_):= if(isconstant(d),d*t^p/p!, if(hasnot(d,t) and has(d,x),psolution(c/b,d/b,y,x,p), 0 )); #psolution(b_,m_,0,d_,y_,t_,x_,m_):= if(isconstant(b), if(has(d,t-x) or has(d,x-t), if(m==1, if(b==1, d*x, int(d,x,m)/(b-1)), int(d,x,m)/(1+b)), if(has(d,t+x), if(b== -1, d*x, int(d,x,m)/(1+b) )))); psolution(b_,m_,c_,c1_*c2_,y_,t_,x_,p_):=if(isconstant(c1), c1*psolution(b,m,c,c2,y,t,x,p), if(isconstant(c2), c2*psolution(b,m,c,c1,y,t,x,p),0 )); psolution(b_,m_,c_,c1_*c2_*c3_,y_,t_,x_,p_):=if(isconstant(c1), c1*psolution(b,m,c,c2*c3,y,t,x,p), if(isconstant(c2), c2*psolution(b,m,c,c1*c3,y,t,x,p) )); psolution(a_,1,0,tt_*xx_,y_,t_,x_,1):= if(t<>x and isconstant(a) and has(tt,t) and has(xx,x) and hasnot(tt,x) and hasnot(xx,t), expand(-int(replace(tt,t,t-x/a+z/a)*replace(xx,x,z),z,0,x)/a) ); psolution(a_,1,0,b_*tt_*xx_,y_,t_,x_,1):= if(t<>x and isconstant(a,b) and has(tt,t) and has(xx,x) and hasnot(tt,x) and hasnot(xx,t), expand(-b/a*int(replace(tt,t,t-x/a+z/a)*replace(xx,x,z),z,0,x)) ); #psolution(b_,m_,c_,ax_*t,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, int(ax,x,m)*t/b-if(p==1,1/b/b*int(ax,x,2m),0),-1/2/c*x*exp(t)-1/4/c*exp(t)-exp(t)/2 )); #psolution(b_,m_,c_,ax_*t,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, int(ax,x,m)*t/b-if(p==1,1/b/b*int(ax,x,2m),0), if(p>1, psolution(c/b,ax/b,y,x,m)*t, psolution(c/b,ax/b,y,x,m)*t+c*int(ax/b,x,m)/2 ))); #psolution(b_,m_,c_,tt_*x,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, int(tt,t,p)*x-if(m==1,b*int(tt,t,2p),0), 1/b*exp(c*t) * integrate(exp(-c*t)* tt,t)*x+integrate(tt,t)/b/c/2 )); #psolution(b_,m_,c_,tt_*x,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, int(tt,t,p)*x-if(m==1,b*int(tt,t,2p),0), if(m>1, psolution(c,tt,y,t,p)*x, -b*exp(c*t) * integrate(exp(-c*t)*tt,t)*x ))); #psolution(b_,m_,c_,tt_*x_,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, int(tt,t,p)*x-if(m>1,0,b*int(tt,t,2p)), if(m>1, psolution(c,tt,y,t,p)*x, psolution(c,tt,y,t,p)*x+c*int(tt,t,p)/2 ))); psolution(b_,m_,c_,t_*x_^n_,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, -x^(m+n)*t/b/(n+m)+if(p>1,0,1/b/b*x^(n+2m)/(n+2m)/(n+m) ))); #psolution(b_,m_,c_,exp(t)*ax_,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, -1/b*int(exp(x/b)*ax,x,m)*exp(t+(-1/b)*x), -int(ax,x,m)*exp(t) )); #psolution(b_,m_,c_,exp(t)*xx_,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, -1/b*int(exp(x/b)*xx,x,m)*exp(t+(-1/b)*x), -int(xx,x,m)*exp(t) )); #psolution(b_,m_,c_,exp(t_)*ax_,y_,t_,x_,p_):= if(isconstant(b,c), psolution(-1/b+c/b,ax/b,y,x,m)*exp(t) ); #psolution(b_,m_,c_,exp(t_)*x_,y_,t_,x_,p_):= if(isconstant(b,c), psolution(-1/b+c/b,x/b,y,x,m)*exp(t) ); #psolution(b_,1,c_,exp(t)*x^n_,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, -(-b)^(1+n)*Gamma(1+n,(-1/b)*x)*exp(t+(-1/b)*x),-(-b/2)^(1+n)*Gamma(1+n,(-2/b)*x)*exp(t+(-2/b)*x) )); #psolution(b_,m_,c_,exp(x)*at_,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, int(exp(b*t)*at,t,p)*exp((-b)*t+x), int(at,t,p)*exp(x) )); #psolution(b_,m_,c_,exp(x)*tt_,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, int(exp(b*t)*tt,t,p)*exp((-b)*t+x), int(tt,t,p)*exp(x) )); #psolution(b_,m_,c_,exp(x_)*at_,y_,t_,x_,p_):= if(isconstant(b,c) or c-b==0, -psolution(-b+c,at,y,t,p)*exp(x) ); psolution(b_,m_,c_,exp(x_)*t_,y_,t_,x_,p_):= if(isconstant(b,c) or c-b==0, -psolution(-b+c,t,y,t,p)*exp(x) ); #psolution(b_,m_,c_,exp(x)*t,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, 1/b*(if(p>1,0,if(p==1,-1/b,1/b-t^p/p!))+t)*exp(x),if(b==c, exp(x)*t*t/2,exp(x)*(-c*t+b*t-if(p>1,0,if(p==1,1,-p+t^p/p! )))/(b-c)^2 ))); #psolution(b_,m_,c_,exp(t)*x,y_,t_,x_,p_):= if(isconstant(b,c), if(c==0, (if(m>1,0,if(m==1,-b,1/b-x^m/m!))+x)*exp(t),if(c==1,exp(t)/2*if(m>1,t*x,(2t*x-t*t)),exp(t)*((1-c)*x-if(m>1,0,if(m==1,b,-m+x^m/m! )))/(1-c)^2 ))); #psolution(b_,m_,0,exp(x_)*t_,y_,t_,x_,p_):= if(isconstant(b) and t<>x, (1/b-t)*exp(x),0 ); psolution(b_,m_,0,exp(t_)*x_,y_,t_,x_,p_):= if(isconstant(b) and t<>x, (b-x)*exp(t),0 ); psolution(b_,m_,c_,exp(t_+x_),y_,t_,x_,p_):= if(isconstant(b,c) and t<>x, if(1+b+c==0, -exp(t+x)*x, -exp(t+x)/(1+b+c) ),0); psolution(b_,m_,c_,exp(t_-x_),y_,t_,x_,p_):= if(isconstant(b,c) and t<>x, if(1-b+c==0 and p==1, -exp(t-x)*x/b, -exp(t-x)/(1+(-1)^m*b+c) ),0); psolution(b_,m_,c_,t_*x_,y_,t_,x_,p_):= if(isconstant(b,c), if(m>1, if(p>1, -1/c*t*x, psolution(c,t,y,t,p)*x), if(p>1, psolution(c,x,y,x,m)*t/b, -(1+c*t)*x/c^2-(2+c*t)*b/c^3 ))); psolution(b_,m_,0,t_*x_,y_,t_,x_,p_):= if(isconstant(b), if(p>1, if(m>1, -1/2*t^(1+p)/(1+p)!*x+1/2*x^(1+m)/(1+m)!*t/b, -t/b*x^(1+m)/(1+m)!+if(p>1,0,int(x,x,2m)/b^2 )), -t^(1+p)/(1+p)!*x+if(m>1,0,b*int(t,t,2p)) )); #psolution(b_,m_,c_,t*x,y_,t_,x_,p_):= if(isconstant(b,c), if(p>1, psolution(c/b,x/b,y,x,m)*t, if(m==1, if(c==0, int(t,t,p)*x-b*int(t,t,2p), -1/(c)*t*x-b/c/c*t-1/c/c*x+b*if(abs(c)==1,-2sgn(c),-sgn(c)/c/c) ), int(x,x,m)*t/b-1/b/b*int(x,x,2m) ))); #psolution(b_,m_,c_,t*x,y_,t_,x_,p_):= if(isconstant(b,c), if(m>1, if(p>1, -1/c*t*x, psolution(c,t,y,t,p)*x), if(p>1, psolution(c,x,y,x,m)*t/b, 1/c*t*x+psolution(c,t,y,t,p)*x+psolution(c,x/b,y,x,m)*t-b*if(abs(c)==1,2sgn(c),sgn(c)/c/c) ))); #psolution(b_,m_,0,t*x,y_,t_,x_,p_):= if(isconstant(b), if(p>1, if(m>1, 1/2*int(t,t,p)*x+1/2*int(x,x,m)*t/b, psolution(0,x/b,y,x,m)*t), if(p<1,int(t,t,p)*x-b*int(t,t,2p), if(m<1,int(x,x,m)*t/b-int(x,x,2m)/b^2 )))); #psolution(b_,m_,0,t_*x_,y_,t_,x_,1):= if(isconstant(b), if(m>1, b*t^2*x/2,int(x,x,m)*t/b-int(x,x,2m)/b^2 )); #psolution(b_,1,0,t_*x_,y_,t_,x_,p_):= if(isconstant(b), if(p>1, x^2*t/2/b,int(t,t,p)*x-b*int(t,t,2p) )); #psolution(b_,p_,0,t_*x_,y_,t_,x_,p_):= if(isconstant(b), int(t,t,p)*x/2-int(x,x,p)*t/2/b+if(p>1,0, -int(t,t,p)/2+int(x,x,p)/2/b )); psolution(b_,1,0,1/t_*x_,y_,t_,x_,1):= if(isconstant(b),(b*t-x)*log(b*t)-x); psolution(b_,1,0,t_*1/x_,y_,t_,x_,1):= if(isconstant(b),(x-b*t)*log(x)/b^2-x/b^2); psolution(b_*t_,1,0,x_,y_,t_,x_,1):=if(isconstant(b), b*t^3/3-t*x ); psolution(t_,1,0,t_*x_,y_,t_,x_,1):= x^2/2; psolution(x_,1,0,t_*x_,y_,t_,x_,1):= x-t*x; psolution(t_,1,0,x_,y_,t_,x_,1):= t^3/3-t*x; #psolution(b_,m_,c_,c1_+c2_,y_,t_,x_,n_):= if(isconstant(c1),c1*t^n/n!+psolution(b,m,c,c2,y,t,x,n), if(has(c2,x) and hasnot(c2,t), psolution(c/b,c2/b,y,x,m)+if(has(c1,t) and hasnot(c1,x), psolution(c,c1,y,t,n), psolution(b,m,c,c1,y,t,x,n) ), if(has(c2,t) and hasnot(c2,x), psolution(c,c2,y,t,n)+if(has(c1,x) and hasnot(c1,t), psolution(c/b,c1/b,y,x,m),psolution(b,m,c,c1,y,t,x,n) ), if(has(c1,t) and hasnot(c1,x), psolution(c,c1,y,t,n)+if(has(c2,x) and hasnot(c2,t), psolution(c/b,c2/b,y,x,m),psolution(b,m,c,c2,y,t,x,n) ), if(has(c1,x) and hasnot(c1,t), psolution(c/b,c1/b,y,x,m)+if(has(c2,t) and hasnot(c2,x), psolution(c,c2,y,t,n),psolution(b,m,c,c2,y,t,x,n) ), psolution(b,m,c,c1,y,t,x,n)+psolution(b,m,c,c2,y,t,x,n) ))))); psolution(b_,m_,c_,c1_+c2_,y_,t_,x_,n_):= psolution(b,m,c,c1,y,t,x,n)+psolution(b,m,c,c2,y,t,x,n);