Wolfram Cloud Document

In[]:=

CompoundExpression[

]

deploy

Thu 27 Apr 2023 16:21:21

Simple failing example

In[]:=

ClearAll["Global`*"];h=(i+1)^-2;inti[expr_]:=Integrate[expr,{i,0,Infinity}];eqn=D[f[i,t],t]==hf[i,t]+hinti[hf[i,t]];sol=DSolveValue[{eqn,f[i,0]==1},f[i,t],{i,t}]f0=Function@@{{i,t},sol};Print["Solution: ",f0[i,t]];Print["Back-substitute: ",eqn0=eqn/.f->f0];Print["Solution correct: ",eqn0/.{i->2,t->3}](*False*)

Out[]=

2t

1+2i+

2

i



Solution:

2t

1+2i+

2

i



Back-substitute:

2

2t

1+2i+

2

i



1+2i+

2

i



2t

1+2i+

2

i



2

(1+i)

+

π

2

Erfi[

2

t

]

2

(1+i)

t

Solution correct: False

Normalized version

In[]:=

ClearAll["Global`*"];$Assumptions={p>0};solve[h2_]:=z=

∞

∫

0

h2i;h=h2z;eqn=D[et[i,t],t]==ahet[i,t]+bh*

∞

∫

0

et[i,t]hi;initialValue=et[i,0]==1;sol=DSolveValue[{eqn,initialValue},et[i,t],{{i,0,∞},{t,0,∞}}];SF=StringForm;simple[expr_]:=HoldForm[ht]Simplify

1

ht

PowerExpand@Log@expr;format[source_,target_]:=Print@SF["solution for h=``: ``",source,target];Block[{h=(i+1)^-(p+1)},format[h,simple@solve[h]]];Block[{h=Exp[-i]},format[h,simple@solve[h]]];Block[{h=Exp[-11i]},format[h,simple@solve[h]]];Block[{h=Exp[-

2

i

]},format[h,simple@solve[h]]];Blockh=Exp[-

i

],format[h,simple@solve[h]];Block[h=Exp[-

1/3

i

],format[h,simple@solve[h]]];Block[{h=Exp[-i]Sin[i]},format[h,simple@solve[h]]];

solution for h=

-1-p

(1+i)

: (a+b)(ht)

solution for h=

-i



: (a+b)(ht)

solution for h=

-11i



: (a+b)(ht)

solution for h=

-

2

i



: (a+b)(ht)

solution for h=

-

i



: (a+b)(ht)

solution for h=

-

1/3

i



: (a+b)(ht)

solution for h=

-i



Sin[i]: (a+b)(ht)

Unnormalized version

In[]:=

ClearAll["Global`*"];$Assumptions={p>0};solve[h_]:=eqn=D[et[i,t],t]==ahet[i,t]+bh*

∞

∫

0

et[i,t]hi;initialValue=et[i,0]==1;sol=DSolveValue[{eqn,initialValue},et[i,t],{{i,0,∞},{t,0,∞}}];SF=StringForm;simple[expr_]:=HoldForm[ht]Simplify

1

ht

PowerExpand@Log@expr;format[source_,target_]:=Print@SF["solution for h=``: ``",source,target];Block[{h=(i+1)^-(p+1)},format[h,simple@solve[h]]];Block[{h=Exp[-i]},format[h,simple@solve[h]]];Block[{h=Exp[-11i]},format[h,simple@solve[h]]];Block[{h=Exp[-

2

i

]},format[h,simple@solve[h]]];Blockh=Exp[-

i

],format[h,simple@solve[h]];Block[h=Exp[-

1/3

i

],format[h,simple@solve[h]]];Block[{h=Exp[-i]Sin[i]},format[h,simple@solve[h]]];

solution for h=

-1-p

(1+i)

: a+

b

p

(ht)

solution for h=

-i



: (a+b)(ht)

solution for h=

-11i



: a+

b

11

(ht)

solution for h=

-

2

i



: a+

b

π

2

(ht)

solution for h=

-

i



: (a+2b)(ht)

solution for h=

-

1/3

i



: (a+6b)(ht)

solution for h=

-i



Sin[i]: a+

b

2

(ht)