Solving differential equation with a constant

[zarei]

Hi,
I am working with mathematica. I have a simple differential equation as follows:
y''[t]+k^2y[t]+(1/t^2)y[t]=0
where k is a constant between 1<k<20.
How can solve this equation and then plot y in terms of k?
thanks

 

[antibrane]

You can solve it using

DSolve[y''[t] + k^2 y[t] + (1/t^2) y[t] == 0, y[t], t]

which gives the answer in terms of Bessel functions. To plot it though, you will need to choose some initial conditions, which you would insert like

DSolve[{y''[t] + k^2 y[t] + (1/t^2) y[t] == 0,y[0]==0,y'[0]==0}, y[t], t]

and to plot with $k$ on the horizontal axis you will need to choose $t$ (or just make a three-dimensional plot). The plot may be done like

sol=DSolve[{y''[t] + k^2 y[t] + (1/t^2) y[t] == 0,y[0]==0,y'[0]==0}, y[t], t]
Plot[y[t]/.sol/.t->t0 , {k,1,20}]

where $t0$ is the $t$ you are choosing.

 

[Mute]

sol=DSolve[{y''[t] + k^2 y[t] + (1/t^2) y[t] == 0,y[0]==0,y'[0]==0}, y[t], t]
Plot[y[t]/.sol/.t->t0 , {k,1,20}]

Just a note: if you choose y(0), y'(0) = 0, the unique solution for this initial condition is y(t) = 0, so I suggest choosing something different for the initial conditions!

 

[zarei]

Thanks for reply,
I have a similar equation where cannot be solved with DSolve. It can be solved just by NDSolve. Could you please guide me how the final solution can be plotted in terms of constant k when we use the NDSolve?

 

[blue_raver22]

Hi there,

To solve this differential equation, you can use the built-in function in Mathematica called DSolve. This function allows you to find the general solution for a given differential equation. In your case, the syntax would be:

DSolve[{y''[t] + k^2 y[t] + (1/t^2) y[t] == 0, y[1] == a, y'[1] == b}, y[t], t]

where a and b are arbitrary constants. This will give you the general solution in terms of k.

To plot y in terms of k, you can use the Plot function in Mathematica. The syntax would be:

Plot[y[t] /. sol, {k, 1, 20}]

where sol is the output from the DSolve function. This will give you a plot of y vs. k for the range of k values specified.

I hope this helps! Let me know if you have any further questions.