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plotting - Manipulate a Differential Equation result



I want to Manipulate the result of Differential Equation like :


F[x_] = y[x] /. First@DSolve[x - y'[x] + y''[x] == 0, y[x], x]
Manipulate[Plot[F[x], {x, -10, 10}], {C[1], 1, 6}, {C[2], -2, 5}]

the result of the equation is :


x + x^2/2 + E^x C[1] + C[2]

but I don't get any Curve on the display.


enter image description here



Answer




There are many fixes to this issue. I would recommend formulating problem from the start in terms of your constants. So you know exactly what constants mean.


F[x_, a_, b_] = y[x] 
            /. First@ DSolve[{x - y'[x] + y''[x] == 0, y[0] == a, y'[0] == b}, y[x], x];

Manipulate[Plot[F[x, a, b], {x, -10, 10}, PlotLabel -> F[x, a, b]], 
  {{a, -4, "initial function"}, -10, 10, Appearance -> "Labeled"}, 
  {{b, .96, "initial 1st derivative"}, .5, 1.5, Appearance -> "Labeled"}]

enter image description here


For your more complicated case mentioned in the comments do this:



G[x_, a_, b_, c_] = {y[x], z[x]} /. First@DSolve[{y'[x] - 8*z'[x] == x^2, 
  z''[x] == x - y[x], y[0] == a, y'[0] == b, z[0] == c}, {y[x], z[x]}, x] // FullSimplify;

G[x, a, b, c] // Column // TraditionalForm

enter image description here


Manipulate[Plot[Evaluate@G[x, a, b, c], {x, -5, 5}, Filling -> 0, 
PlotLabel -> Column[G[x, a, b, c]]], 
{{a, 8, "initial y"}, -10, 10,Appearance -> "Labeled"}, 
{{b, 0, "initial y'"}, -10, 10, Appearance -> "Labeled"}, 

{{c, 0, "initial z"}, -10, 10, Appearance -> "Labeled"}]

enter image description here


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