# Differential Equations : Three Initial-Value Problems

Theorem 8.3. Assume P and Q are continuous on an open interval I. Choose any point a in I and let b be any real number. Then there is one and only one function y = f(x) which satisfies the initial value problem

y' + P(x)y = Q(x), with f(a) = b on the interval I. This function is given by the formula

f (x) = be- A( x) + e-A( x) ∫Q(t)eA(t )dt

Where .......

Please provide solutions to the following three problems (using the above Theorem):

.....

Prove that there is exactly one function f, continuous on the positive real axis, such that

.....

for all x > 0 and find this function.

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Problem #3

, , , , , .

Then

According to the Theorem 8.3, ...

#### Solution Summary

Initial value problems using differential equations are solved. The solution is detailed and well presented.