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First order homogeneous LTI Differential Equations
There are several methods for solving a differential equation. These are in time domain and in an appropriate transformed domain such as s-domain or frequency domain. For linear time-invariant (LTI) differential equations, only, the method of Laplace transformation can be used that converts the differential equations into a set of algebraic equations, which is much easier to solve. The solution is of course in s-domain instead of time domain and needs to be converted back to time domain using inverse Laplace transformation.
In the following we consider a direct method of solution for differential equations in its original time domain.
Case 1
In this case the first derivative of a function y(t) (i.e., ) is expressed in terms of an exogenous function f(t), which is independent of y(t) and its derivatives.
That is
where dy(t)/dt is the first derivative of y(t)
To solve for y(t), one may integrate both sides of the equation with respect to t. Then y(t) can be found as:
Now it can be seen that:
where c = y(t0) is a constant, due to initial conditions.
Example
 , c=y(t0)
 , c1 = y(t0) + cos(t0)
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