Solving an ODE containing a function of the independent variable only known at discrete points

I'm a little new at Mathematica and I'm trying to solve the following ordinary differential equation:

$qquad xfracdy(x)dx+y=-p(x)$.

I have data (in a list) for $x$, and the corresponding values of $p(x)$ (also in a list). I need to solve the ODE to get a list $y(x)$ for the $x$s in the 1st list. I've been trying to use NDSolve, like so:

NDSolve

NDSolve[x*y'[x] + y[x] == -p, y, x, a, b]


with no luck.

Is this possible in Mathematica? Is there a way to do this?

p

y

2 Answers
2

Why not try to fit p(x) to the data and then using DSolve? Since you did not make a MWE, I made up some data

p(x)

DSolve

ClearAll[x, y, p]
xData = 1, 2, 3, 4;
pData = 1, 4, 9, 16;
data = Transpose[xData, pData];
p[x_] = Fit[data, 1, x, x^2, x]; (*change fit as needed*)
DSolve[x y'[x] + y[x] == -p[x], y[x], x]


Interpolation

Fit

It is a very simple equation and I recommend to first solve it analytically. It can be done by the, say, the so-called, u*v method. The solution is as follows:

Now one can directly integrate it numerically by summing up the areas of the trapezoids formed by the points x1,x2,p1 and p2. Alternatively, one can use the advice of @Nasser and integrate the fitted function, which is easier.

Have fun!

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