# Differential Equations

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.

https://brainmass.com/math/calculus-and-analysis/differential-equations-reduction-order-methods-6190

#### Solution Preview

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). The ...

#### Solution Summary

Two differential equations are solved. The differential equations for reduction of order methods are determined. Limits that go to infinity are given.