Lebl-Basic-Analysis-I · Problem 6.3.6

Sequences of Functions

Mathematics · picards-theorem-linear-ode

Problem

A common type of equation one encounters are linear first order differential equations, that is equations of the form

y' + p(x)y = q(x), \quad y(x_0) = y_0.

Prove Picard's theorem for linear equations. Suppose

I

is an interval,

x_0 \in I

, and

p : I \to \mathbb{R}

and

q : I \to \mathbb{R}

are continuous. Show that there exists a unique differentiable

f : I \to \mathbb{R}

, such that

y = f(x)

satisfies the equation and the initial condition. Hint: Assume existence of the exponential function and use the integrating factor formula for existence of

f

(prove that it works and then that it is unique):

f(x) \mathrel{:=} e^{-\int_{x_0}^x p(s)\,ds} \left( \int_{x_0}^x e^{\int_{x_0}^t p(s)\,ds} q(t)\,dt + y_0 \right).

Related theory

The Prime Number Theorem And Dirichlet S Theorem

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