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The IVP sin(t) {d^2x} / {dt^2} + cos(t) {dx} / {dt} + sin (t) x = tan (t), x (0.5) = 18, {dx} /...

Question:

The IVP

{eq}\displaystyle \sin(t) \dfrac {d^2x} {dt^2} + \cos(t) \dfrac {dx} {dt} + \sin (t) x = \tan (t),\ x (0.5) = 18,\ \dfrac {dx} {dt}\Bigg|_{0.5} = 10 {/eq}. Find the interval on which the unique solution of the IVP is defined.

Interval for Existence of Unique Solution of Second Order Initial Value Problem (IVP)


The question presents a second-order, linear, ordinary differential equation (ODE) with a right hand side. Also given are initial conditions (IC) that make up an initial value problem (IVP). Since the ODE is linear, we determine intervals on the real number line where its coefficients are continuous. Then using the Theorem of Existence and Uniqueness, from the theory of ODEs, we find the largest interval where a unique solution of the IVP exists. The concepts used from Calculus include continuity.


Answer and Explanation: 1

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Dividing the ordinary differential equation (ODE) by {eq}\sin t \; (\sin t \ne 0) {/eq} on both sides yields the ODE

{eq}\displaystyle \frac {d^2...

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Existence Proofs in Math: Definition & Examples

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Chapter 3 / Lesson 4
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Examine existence theorems in mathematics. Learn how to construct an existence proof. Study some of the most important existence theorems and see examples.


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