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Differential Equations

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(i) Consider the differential equation:
x. = x^2 , x(0) given x(0)>0

Find the solution of x(t) of this equation in terms of x(0) and show that there is a T, which depends on x(0), such that lim x(t) = infinity t --> T-

(ii) Find the solution of the differential equation
x.. - x = e^-t/2 , x(0) = 0, (0) = 1,
Using the reduction of order method.

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Solution. (1) From x'=x^2, we have
dx/dt=x^2
Then 1/x^2dx=dt.
Integrate both sides to get
-1/x=t+C,
where C is a constant. According to the initial condition we know that, when t=0, x=x(0)>0. So C=-1/x(0). ...

Solution Summary

Two differential equations are solved.

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