cannot be expressed in terms of a finite number of elementary functions. But the value of the integral can be approximated for different values of t.
I am having problems verifying a solution for a differential equation. The equation is
The solution to the equation is
My problem is calculating the integral that is contained on the solution. I thought there was no way to calculate an integral of
at least that is what a tutor told me at the tutoring service the university provides.
Shouldn't there be a constant of integration in there? When I solve for the integrating factor and integrate I get:
Now differentiate :
Now you can back-substitute that into the differential equation to verify the solution. Also, just for fun, you can work with it in Mathematica: define y(t), y(0)=a, differentiate it, substitute it back into the DE:
Code:In[73]:= y[t_] := Exp[t^2]*Integrate[Exp[-s^2], {s, 0, t}] + a*Exp[t^2] FullSimplify[D[y[t], t] - 2*t*y[t]] Out[74]= 1
The differential equation in the OP is a linear one. The standard solution to this is given in any textbook on differential equations. The resulting solution I got for this equation is:
So, the integration constant is K and the derivative of the integral can easily be found to become the DE again. I do not see why one should have a definite integral in the solution.
coomast