MATLAB: How to numerically solve a differential equation with a dirac delta function

Question

The differential equation that I want to solve is

Upon using ode45 and the dirac function, the dirac function doesn't seem to have any effect (which makes sense because x never reaches 1 in a numerical solution)

Any ideas on how to solve this numerically?

Answer 1

Use ode45 to solve the equation over the start time to time 1 (the place the dirac delta is located.) Do not use the if statements like Alan and Mohit show: just end the integration at the point they would take effect. Using if presents theory problems that you can avoid by just stopping at the place of the event.

Now take the final output of that ode45 and give the appropriate kick to the boundary conditions.

Then restart the ode45 from time 1 to the final desired time, passing in the kicked boundary conditions. Do not use if -- again you avoid the theory problem by not having ode45 cross the interval of discontinuity.