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Mathematics Calculus Equation Solving Wolfram Language

Get solution that satisfies the equation?

Jacques Ou

Posted 9 years ago

The codes are eq12 = (cinf \[Theta]c \[Lambda])/(s + \[Theta]c) - ( E^((lh - x) /Sqrt[Dc ] Sqrt[s + \[Theta]c]) F1 Sqrt[ Dc ] \[Lambda])/ Sqrt[ (s + \[Theta]c)] /. {Sqrt[Dc] F1 \[Lambda] -> A1, cinf \[Theta]c \[Lambda] -> A2} eq13 = s q[x, s] - DpD[q[x, s], {x, 2}] - pinf + eq12 == 0 eq20 = Assuming[(lh - x) < 0 && s > 0 && s + \[Theta]c > 0 && Dp > 0 && Dc > 0 && \[Alpha]1 > 0, DSolve[eq13, q[x, s], {x, s}, GeneratedParameters -> B1 ]] // Simplify // Normal eq23 = FullSimplify[ eq13 /. eq20[[1]], (lh - x) < 0 && s > 0 && s + \[Theta]c > 0 && Dp > 0 && Dc > 0 && \[Alpha]1 > 0] // Normal eq23 should show 'true'. What's wrong?

The codes are

eq12 = (cinf \[Theta]c \[Lambda])/(s + \[Theta]c) - (
   E^((lh - x) /Sqrt[Dc ] Sqrt[s + \[Theta]c]) F1 Sqrt[
    Dc ] \[Lambda])/
   Sqrt[ (s + \[Theta]c)] /. {Sqrt[Dc] F1 \[Lambda] -> A1, 
   cinf \[Theta]c \[Lambda] -> A2}

eq13 = s *q[x, s] - Dp*D[q[x, s], {x, 2}] - pinf + eq12 == 0

eq20 = Assuming[(lh - x) < 0 && s > 0 && s + \[Theta]c > 0 && Dp > 0 &&
      Dc > 0 && \[Alpha]1 > 0, 
    DSolve[eq13, q[x, s], {x, s}, GeneratedParameters -> B1 ]] // 
   Simplify // Normal

eq23 = FullSimplify[
   eq13 /. eq20[[1]], (lh - x) < 0 && s > 0 && s + \[Theta]c > 0 && 
    Dp > 0 && Dc > 0 && \[Alpha]1 > 0] // Normal

eq23 should show 'true'. What's wrong?

POSTED BY: Jacques Ou

4 Replies

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Jacques Ou

Posted 9 years ago

Thanks.

POSTED BY: Jacques Ou

Michael Rogers

Michael Rogers, Emory University

Posted 9 years ago

POSTED BY: Michael Rogers

Jacques Ou

Posted 9 years ago

Hello, Mr. Rogers, Using DSolve[eq13, q, {x, s}, GeneratedParameters -> B1] will produce a pure function and the following calculation will be based on function. But using DSolve[eq13, q[x,s], {x, s}, GeneratedParameters -> B1] will produce an expression and the following calculation will be based on substitution. Is it correct? So if I redefine q[x,s] and it can work likewise. In[45]:= q[x_, s_] := -(E^(-((Sqrt[s] x)/Sqrt[Dp]) - Sqrt[s/Dp] (lh + x)) (-(E^( Sqrt[s/Dp] (2 lh + x)) (-Sqrt[s] + Sqrt[Dp] (Sqrt[s/Dp] - \[Alpha]1)) + E^(Sqrt[s/Dp] (lh + 2 x)) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) + E^(lh Sqrt[s/Dp] + (2 Sqrt[s] x)/Sqrt[ Dp]) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) + E^((2 lh Sqrt[s])/Sqrt[Dp] + Sqrt[s/Dp] x) (Sqrt[s] - Sqrt[Dp] (Sqrt[s/Dp] + \[Alpha]1))) Sqrt[ Dp s (s + \[Theta]c)] (-Dc s + Dp (s + \[Theta]c)) (-A2 + pinf (s + \[Theta]c)) - A1 Sqrt[Dp s] (s + \[Theta]c) (Dp^( 3/2) (E^(Sqrt[s/Dp] (2 lh + x)) - E^((2 lh Sqrt[s])/Sqrt[Dp] + Sqrt[s/Dp] x)) Sqrt[s/ Dp] (s + \[Theta]c) + Dc (E^(lh Sqrt[s/Dp] + (2 Sqrt[s] x)/Sqrt[ Dp] + (lh - x) Sqrt[(s + \[Theta]c)/Dc]) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (-s + Dp Sqrt[s/Dp] Sqrt[(s + \[Theta]c)/Dc]) - E^(lh Sqrt[s/Dp] + 2 Sqrt[s/Dp] x + (lh - x) Sqrt[(s + \[Theta]c)/ Dc]) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + Dp Sqrt[s/Dp] Sqrt[(s + \[Theta]c)/Dc]) + E^((2 lh Sqrt[s])/Sqrt[Dp] + Sqrt[s/Dp] x) (-s^(3/2) + Sqrt[Dp] s (Sqrt[s/Dp] + \[Alpha]1) + Dp Sqrt[s] Sqrt[s/Dp] Sqrt[(s + \[Theta]c)/Dc] - Dp^(3/2) Sqrt[s/Dp] \[Alpha]1 Sqrt[(s + \[Theta]c)/ Dc]) + E^( Sqrt[s/Dp] (2 lh + x)) (s^(3/2) + Sqrt[Dp] s (-Sqrt[(s/Dp)] + \[Alpha]1) + Dp Sqrt[s] Sqrt[s/Dp] Sqrt[(s + \[Theta]c)/Dc] + Dp^(3/2) Sqrt[s/Dp] \[Alpha]1 Sqrt[(s + \[Theta]c)/ Dc]))) - 2 Dp E^(Sqrt[s/Dp] (lh + x)) s Sqrt[s/ Dp] (E^((2 lh Sqrt[s])/Sqrt[ Dp]) (Sqrt[s] - Sqrt[Dp] \[Alpha]1) + E^((2 Sqrt[s] x)/Sqrt[ Dp]) (Sqrt[s] + Sqrt[Dp] \[Alpha]1)) (s + \[Theta]c)^( 3/2) (-Dc s + Dp (s + \[Theta]c)) B2[1][s]))/(2 Dp^2 (s/Dp)^( 3/2) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c)^( 3/2) (-Dc s + Dp (s + \[Theta]c))) In[46]:= eq34 = FullSimplify[ eq14, (lh - x) < 0 && s > 0 && s + \[Theta]c > 0 && Dp > 0 &&

Hello, Mr. Rogers, Using DSolve[eq13, q, {x, s}, GeneratedParameters -> B1] will produce a pure function and the following calculation will be based on function. But using DSolve[eq13, q[x,s], {x, s}, GeneratedParameters -> B1] will produce an expression and the following calculation will be based on substitution. Is it correct?

So if I redefine q[x,s] and it can work likewise.

In[45]:= q[x_, 
  s_] := -(E^(-((Sqrt[s] x)/Sqrt[Dp]) - 
      Sqrt[s/Dp] (lh + 
         x)) (-(E^(
            Sqrt[s/Dp] (2 lh + x)) (-Sqrt[s] + 
              Sqrt[Dp] (Sqrt[s/Dp] - \[Alpha]1)) + 
           E^(Sqrt[s/Dp] (lh + 2 x)) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) + 
           E^(lh Sqrt[s/Dp] + (2 Sqrt[s] x)/Sqrt[
             Dp]) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) + 
           E^((2 lh Sqrt[s])/Sqrt[Dp] + 
             Sqrt[s/Dp] x) (Sqrt[s] - 
              Sqrt[Dp] (Sqrt[s/Dp] + \[Alpha]1))) Sqrt[
        Dp s (s + \[Theta]c)] (-Dc s + Dp (s + \[Theta]c)) (-A2 + 
          pinf (s + \[Theta]c)) - 
       A1 Sqrt[Dp s] (s + \[Theta]c) (Dp^(
           3/2) (E^(Sqrt[s/Dp] (2 lh + x)) - 
             E^((2 lh Sqrt[s])/Sqrt[Dp] + Sqrt[s/Dp] x)) Sqrt[s/
           Dp] (s + \[Theta]c) + 
          Dc (E^(lh Sqrt[s/Dp] + (2 Sqrt[s] x)/Sqrt[
               Dp] + (lh - x) Sqrt[(s + \[Theta]c)/Dc]) (Sqrt[s] + 
                Sqrt[Dp] \[Alpha]1) (-s + 
                Dp Sqrt[s/Dp] Sqrt[(s + \[Theta]c)/Dc]) - 
             E^(lh Sqrt[s/Dp] + 
               2 Sqrt[s/Dp] x + (lh - x) Sqrt[(s + \[Theta]c)/
                Dc]) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + 
                Dp Sqrt[s/Dp] Sqrt[(s + \[Theta]c)/Dc]) + 

             E^((2 lh Sqrt[s])/Sqrt[Dp] + 
               Sqrt[s/Dp] x) (-s^(3/2) + 
                Sqrt[Dp] s (Sqrt[s/Dp] + \[Alpha]1) + 
                Dp Sqrt[s] Sqrt[s/Dp] Sqrt[(s + \[Theta]c)/Dc] - 
                Dp^(3/2) Sqrt[s/Dp] \[Alpha]1 Sqrt[(s + \[Theta]c)/
                 Dc]) + E^(
              Sqrt[s/Dp] (2 lh + x)) (s^(3/2) + 
                Sqrt[Dp] s (-Sqrt[(s/Dp)] + \[Alpha]1) + 
                Dp Sqrt[s] Sqrt[s/Dp] Sqrt[(s + \[Theta]c)/Dc] + 
                Dp^(3/2) Sqrt[s/Dp] \[Alpha]1 Sqrt[(s + \[Theta]c)/
                 Dc]))) - 
       2 Dp E^(Sqrt[s/Dp] (lh + x)) s Sqrt[s/
        Dp] (E^((2 lh Sqrt[s])/Sqrt[
           Dp]) (Sqrt[s] - Sqrt[Dp] \[Alpha]1) + 
          E^((2 Sqrt[s] x)/Sqrt[
           Dp]) (Sqrt[s] + Sqrt[Dp] \[Alpha]1)) (s + \[Theta]c)^(
        3/2) (-Dc s + Dp (s + \[Theta]c)) B2[1][s]))/(2 Dp^2 (s/Dp)^(
    3/2) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c)^(
    3/2) (-Dc s + Dp (s + \[Theta]c)))

In[46]:= eq34 = 
 FullSimplify[
   eq14, (lh - x) < 0 && s > 0 && s + \[Theta]c > 0 && Dp > 0 &&

POSTED BY: Jacques Ou

Michael Rogers

Michael Rogers, Emory University

Posted 9 years ago

Solve the DE for `q`, not for `q[x,s]`: DSolve[eq13, q, {x, s}, GeneratedParameters -> B1] The solution will yield a function so that the `q` in the derivative `Derivative[2, 0][q][x, s]` in `eq13` will be replaced with the function. Then all works well. Your original form results in a solution `q[x, s] -> ...`, which replaces only `q[x, s]` literally and does not replace its derivatives.

POSTED BY: Michael Rogers

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