A directly integrable first-order ode has the form:
where g(t) is a given function. The equation states that the derivative of the unknown function y(t) is a known function, g(t) and that the unknown function satifies the initial condition y(t_0)=y_0.
Solution Procecure
To compute y(t) we apply the Fundamental Theorem of Calculus. The general solution of the differential equation is
The term C is a constant. C can be determined from an initial condition, if one is given. We can also express the solution in terms of a definite integral. Applying the Fundamental Theorem of Calculus, we have
Example
ODE:
Solution:
Evaluating the integral, we have
The final answer is
Alternatively, we can write the solution as an indefinite integral
Applying the initial condition y(7)=4, we have
This implies C=669.75, and we end up with the same answer as in the above case with the definite integral.
Example
ODE:
Solution:
We cannot simplify the above expression any further since the antiderivative of
cannot be expressed in terms of elementary functions.
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