Example: Find the solution to the system
under the initial condition .
Answer: Notice that the second equation of the system is a differential equation involving only the variable y. Its integration gives
Note that is the derivative of not its anti-derivative! The initial condition translates into the initial condition y(0)=0 for the variable y. Hence, we have
which gives . Since we have y, we plug it into the first equation to get
We recognize a first order linear differential equation. In order to solve it, first we need to find the integrating factor given by
Note that the anti-derivative of is . The general solution is then given by
where, in the first integral, we used direct tables and for the second one we used integration by parts (we integrated and differentiated t). Putting everything together, we get
The initial condition translates into the initial condition x(0)=1 for the variable x. Hence, we have
which gives C=0. Therefore, we have
Finally, the solution to the system is
Note that since , we may generate another expression for the function x(t).
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Author: Mohamed Amine Khamsi