# Change the lower limit of the integral when using DSolve?

Posted 6 months ago
538 Views
|
2 Replies
|
2 Total Likes
|
 Please tell me how to make the integral start from 0 rather than 1 in the result? Thanks. In[1702]:= eq71 = DSolve[{f2[k, t] - Dp*w2[k, t] \[Eta][k] == \!$$\*SuperscriptBox[\(w2$$, \* TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None]\)[k, t], f3[k, t] - Dp*w3[k, t] \[Eta][k] == \!$$\*SuperscriptBox[\(w3$$, \* TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None]\)[k, t]}, {w2[k, t], w3[k, t]}, t] // Simplify // Normal Out[1702]= {{w2[k, t] -> E^(-Dp t \[Eta][k]) (C[1] + \!$$\*SubsuperscriptBox[\(\[Integral]$$, $$1$$, $$t$$]$$\( \*SuperscriptBox[\(E$$, $$Dp\ K[1]\ \[Eta][k]$$]\ f2[k, K[1]]\) \[DifferentialD]K[1]\)\)), w3[k, t] -> E^(-Dp t \[Eta][k]) (C[2] + \!$$\*SubsuperscriptBox[\(\[Integral]$$, $$1$$, $$t$$]$$\( \*SuperscriptBox[\(E$$, $$Dp\ K[2]\ \[Eta][k]$$]\ f3[k, K[2]]\) \[DifferentialD]K[2]\)\))}} 
2 Replies
Sort By:
Posted 6 months ago
 Possible You must do it manually typing.Maybe there is a another solution?  sol = DSolve[{f2[k, t] - Dp*w2[k, t]*\[Eta][k] == D[w2[k, t], t], f3[k, t] - Dp*w3[k, t]*\[Eta][k] == D[w3[k, t], t]}, {w2[k, t], w3[k, t]}, t] For first one:  w2[k, t] /. sol[[1]] With substitution:  K[1]-1==K[3] differentiation on both sides:  \[DifferentialD]K[1] == \[DifferentialD]K[3] and substitution to equation. The second equation the same method:
 I think the problem can be solved by giving initial conditions. In[2345]:= eq71 = DSolve[{f2[k, t] - Dp*w2[k, t] \[Eta][k] == \!$$\*SuperscriptBox[\(w2$$, \* TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None]\)[k, t], f3[k, t] - Dp*w3[k, t] \[Eta][k] == \!$$\*SuperscriptBox[\(w3$$, \* TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None]\)[k, t], w2[k, 0] == eq71, w3[k, 0] == eq72}, {w2[k, t], w3[k, t]}, t] // Simplify // Normal Out[2345]= {{w2[k, t] -> (1/((lh - rb) Sqrt[\[Eta][k]])) E^(-Dp t \[Eta][ k]) (2 pinf (Sin[lh Sqrt[\[Eta][k]]] - Sin[rb Sqrt[\[Eta][k]]]) + (-lh + rb) (\!$$\*SubsuperscriptBox[\(\[Integral]$$, $$1$$, $$0$$]$$\( \*SuperscriptBox[\(E$$, $$Dp\ K[1]\ \[Eta][k]$$]\ f2[k, K[1]]\) \[DifferentialD]K[1]\)\)) Sqrt[\[Eta][ k]] + (lh - rb) (\!$$\*SubsuperscriptBox[\(\[Integral]$$, $$1$$, $$t$$]$$\( \*SuperscriptBox[\(E$$, $$Dp\ K[1]\ \[Eta][k]$$]\ f2[k, K[1]]\) \[DifferentialD]K[1]\)\)) Sqrt[\[Eta][k]]), w3[k, t] -> (1/((lh - rb) Sqrt[\[Eta][k]])) E^(-Dp t \[Eta][ k]) (2 pinf (-Cos[lh Sqrt[\[Eta][k]]] + Cos[rb Sqrt[\[Eta][k]]]) + (-lh + rb) (\!$$\*SubsuperscriptBox[\(\[Integral]$$, $$1$$, $$0$$]$$\( \*SuperscriptBox[\(E$$, $$Dp\ K[2]\ \[Eta][k]$$]\ f3[k, K[2]]\) \[DifferentialD]K[2]\)\)) Sqrt[\[Eta][ k]] + (lh - rb) (\!$$\*SubsuperscriptBox[\(\[Integral]$$, $$1$$, $$t$$]$$\( \*SuperscriptBox[\(E$$, $$Dp\ K[2]\ \[Eta][k]$$]\ f3[k, K[2]]\) \[DifferentialD]K[2]\)\)) Sqrt[\[Eta][k]])}}