Please see the attached file for the fully formatted problems.
(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.
Solution. (1) From x'=x^2, we have
Integrate both sides to get
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). ...
Two differential equations are solved.