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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.

Please see the attached file for the fully formatted problems.

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IVPs are solved. The solution is detailed and well presented.

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