Plot an Integral of NDSolve solution

Question

I have a problem with plotting the integral of an NDSolve solution.

I have attached the code for a simple 2nd order linear ODE to be solved by NDSolve and then to be integrated by NIntegrate.

I would you appreciate if you could give me a help to overcome the problem.

My best regards.

Hadi

f := NDSolve[{x''[t] + 2 x'[t] + x[t] == 0, x[0] == 1, x'[0] == -4}, 


x, {t, 0, 2}]

g[t_] := NIntegrate[Exp[y*Evaluate[x[t] /. f]], {y, 1, 10}]

Plot[g[t], {t, 0, 2}]

And hear are the errors

NDSolve::dsvar: 0.00004085714285714285` cannot be used as a variable. >>

ReplaceAll::reps: {NDSolve[{x[0.0000408571]+2 (x^[Prime])[0.0000408571]+(x^[Prime][Prime])[0.0000408571]==0,x[0]==1,(x^[Prime])[0]==-4},x,{0.0000408571,0,2}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>

NIntegrate::inumr: The integrand E^(y (x[0.0000408571]/. NDSolve[{x[<<1>>]+Times[<<2>>]+(<<1>>^(<<1>>))[<<1>>]==0,x[0]==1,(x^[Prime])[0]==-4},x,{0.0000408571,0,2}])) has evaluated to non-numerical values for all sampling points in the region with boundaries {{1,10}}. >>

NDSolve::dsvar: 0.00004085714285714285` cannot be used as a variable. >>

ReplaceAll::reps: {NDSolve[{x[0.0000408571]+2 (x^[Prime])[0.0000408571]+(x^[Prime][Prime])[0.0000408571]==0,x[0]==1,(x^[Prime])[0]==-4},x,{0.0000408571,0,2}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>

NIntegrate::inumr: The integrand E^(y (x[0.0000408571]/. NDSolve[{x[<<1>>]+Times[<<2>>]+(<<1>>^(<<1>>))[<<1>>]==0,x[0]==1,(x^[Prime])[0]==-4},x,{0.0000408571,0,2}])) has evaluated to non-numerical values for all sampling points in the region with boundaries {{1.,10.}}. >>

NDSolve::dsvar: 0.04085718367346938` cannot be used as a variable. >>

General::stop: Further output of NDSolve::dsvar will be suppressed during this calculation. >>

ReplaceAll::reps: {NDSolve[{x[0.0408572]+2 (x^[Prime])[0.0408572]+(x^[Prime][Prime])[0.0408572]==0,x[0]==1,(x^[Prime])[0]==-4},x,{0.0408572,0,2}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>

General::stop: Further output of ReplaceAll::reps will be suppressed during this calculation. >>

NIntegrate::inumr: The integrand E^(y (x[0.0408572]/. NDSolve[{x[<<1>>]+Times[<<2>>]+(<<1>>^(<<1>>))[<<1>>]==0,x[0]==1,(x^[Prime])[0]==-4},x,{0.0408572,0,2}])) has evaluated to non-numerical values for all sampling points in the region with boundaries {{1,10}}. >>

General::stop: Further output of NIntegrate::inumr will be suppressed during this calculation. >>

Answer 1

A useful piece of advice. Before putting many commands in the same cell, try to run each one individually to make sure that there are no errors. This time it was the first NDSolve causing something. The following code works

f1 = NDSolve[{x''[t] + 2 x'[t] + x[t] == 0, x[0] == 1, x'[0] == -4}, 
  x, {t, 0, 2}]
Plot[x[t] /. f1, {t, 0, 2}]

And then,the numerical integration and its plot

g[t_] := NIntegrate[Exp[y*Evaluate[x[t] /. f1]], {y, 1, 10}]
Plot[g[t], {t, 0, 2}]

Answer 2

There is no need to use NIntegrate:

X = NDSolveValue[{x''[t] + 2 x'[t] + x[t] == 0, x[0] == 1,x'[0] == -4}, x, {t, 0, 2}]

Plot[Integrate[Exp[y X[t]], {y, 1, 10}],{t,0,2}]

Answer 3

Clear["Global`*"]

eqns = {x''[t] + 2 x'[t] + x[t] == 0, x[0] == 1, x'[0] == -4};

In defining f use Set rather than SetDelayed

f = NDSolve[eqns, x, {t, 0, 2}][[1]]

g[t_?NumericQ] :=
 NIntegrate[Exp[y*x[t] /. f], {y, 1, 10}]

For comparison, the exact solution of the differential equation is

sol = DSolve[eqns, x, t][[1]]

(* {x -> Function[{t}, -E^-t (-1 + 3 t)]} *)

Verifying the exact solution

eqns /. sol // Simplify

(* {True, True, True} *)

And the exact expression for g is

g2[t_] = Integrate[Exp[y*x[t]] /. sol, {y, 1, 10}]

(* (E^(E^-t (1 + (-30 + E^t) t)) (-E^(9 E^-t) + E^(27 E^-t t)))/(-1 + 3 t) *)

g2[t] has a minimum

min = Minimize[g2[t], t] // Simplify

(* {1/3 E^(4/3 - 30/E^(4/3)) (-1 + E^(27/E^(4/3))), {t -> 4/3}} *)

Plot[{g[t], g2[t]}, {t, 0, 2},
 PlotStyle -> {Automatic, Dashed},
 PlotLegends -> Placed[{Numeric, Exact}, {0.5, 0.5}],
 Epilog ->
  {Red, AbsolutePointSize[4],
   Point[{t, g2[t]} /. min[[2]]]}]