# Question for function FindMaximum

I need some advice on using the function FindMaximum as follows.

I find the function y(x) from a differential equation as below, and then for each c and each xMax I have different solutions y(x) and I evaluate Abs(y(x=xMax)).

test[a_, xMax_] := Module[{temp},
temp = NDSolve[{y'[x] == c* y[x] Cos[x + y[x]], y == 1},
y, {x, 0, xMax}];

(y /. temp)[][xMax] // Abs (*Abs(y(x=xMax))*)
]


I fix xMax = 30 and narrow c from 0.2 to 0.5 and I want to find c that gives me the largest Abs(y(x=xMax)).

FindMaximum[{test[c, 30], 0.2 < c < 0.5}, {c, 0.3}]



But the code doesn't work. Errors appear like How can I resolve this.

• Replace test[c_, xMax_] := Module[... with Clear[test];test[c_?NumericQ, xMax_] := Module[... and see if your problem goes away. That Clear[test] removes any cached old definition of test and the ?NumericQ forces evaluation to be done on numeric constants given to c instead of any possible symbolic evaluation of c without an assigned numeric constant value.
– Bill
Dec 11 '19 at 2:23
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• @Bill: Thank you. It works. Dec 11 '19 at 3:28

ParametricNDSolve is well suited for such problems. Try
test = ParametricNDSolveValue[{y'[x] == c*y[x] Cos[x + y[x]], y == 1}, y[xMax],

By the way, the code in the question contains an error. Replace test[a_, xMax_] by test[c_, xMax_].