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

We have

,

and

,

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).

[Differential Equations] [First Order D.E.]
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