Problem Statement (K 2011 p.59 pb. 3)
Find a general solution. Check your answer by substitution.
We can write the above differential equation in the following form:
The characteristic equation of the given DE is
Now, in order to solve for , we can use the quadratic formula:
Therefore, we have:
Thus, we have found that the general solution of the DE is actually:
To check if is indeed the solution of the given DE, we can differentiate the
what we found to be the general solution.
Substituting the values of and in the given equation, we get:
Therefore, the solution of the given DE is in fact:
Problem Statement (K 2011 p.59 pb. 4)
Find a general solution to the given ODE. Check your answer by substituting into the original equation.
The characteristic equation of this ODE is therefore:
Evaluating the discriminant:
Therefore the equation has two complex conjugate roots and a general homogenous solution of the form:
And finally we find the general homogenous solution:
We found that:
Differentiating to obtain and respectively:
Substituting these equations into the original ODE yields:
Therefore, the solution is correct.