{"text_format":"3","workbook":"xlsx","n_sheets":3,"sheets":["Sheet1","Sheet2","Sheet3"],"not_rendered":{"charts":1,"macros":1,"defined_names":9,"comments":1},"read":"whole","date_system":"1900"}

{"sheet":"Sheet1","index":0,"n_rows":41,"n_cols":14,"rows_kept":41,"merged":[[34,2,34,6]]}
[null,"     Orthogonal Trajectories for"]
[null,null,"         y' = f(x, y)"]
[]
[null,"              This DIGMath program illustrates "]
[null,"       the concept of the orthogonal trajectories"]
[null,"        for any first order differential equation"]
[{"$nulls":3},"y' = f(x, y)"]
[null,"             with initial conditions x0 and y0"]
[null,"             extending from x = x0   to x = b."]
[]
["Enter the function in terms of both x and y:"]
[{"$nulls":4}," f(x, y) =","x*(y+1)^3"]
[]
["Now enter the initial conditions:"]
[null,"The initial value x0 of  x is",null,null,1]
[null,"The initial value y0 of  y is",null,null,5]
[]
["To what value of x should the solutions extend?",{"$nulls":7},"       The solutions (in blue) of the D.E.  y' =",{"$nulls":4},{"$f":"F12","v":"x*(y+1)^3"}]
[null,null,"xMax =",5]
[{"$nulls":8},"Select the starting height ",null,null,{"$f":"\"(\"&TEXT(y0,\"#\")&\" - \"&TEXT(y0+6,\"#\")&\")\"","v":"(5 - 11)"},"of middle solution curve  "]
[]
[{"$nulls":9},{"$f":"5+K22*0.25","v":7.5},10]
[]
[]
["Click each item below for suggestions and investigations",{"$nulls":7},"The orthogonal trajectories (in red) "]
[null,"   Item 1"]
[null,"   Item 2",{"$nulls":6},"Select the starting height ",null,null,{"$f":"\"(\"&TEXT(y0,\"#\")&\" - \"&TEXT(y0+8,\"#\")&\")\"","v":"(5 - 13)"},"of middle orthogonal trajectory"]
[null,"   Item 3"]
[null,"   Item 4",{"$nulls":7},{"$f":"5+K29*0.25","v":7.5},10]
[]
[]
[]
[]
["To access/download all the DIGMath Apps for Differential Equations and Matrix Algebra"]
["     Click this link:",null,"DIGMath for Differential Equations/Matrix Algebra"]
[]
["Created by:  Sheldon P. Gordon   ",null,null,"© 2026"]
["                      Farmingdale State College    "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

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