Extraordinary Differential Equations #12

Calculus Level 4

Consider the differential equation on a series of cosine functions involving $y=y(x)$ : $\frac{dy}{dx}=\sum_{k=1}^{2017} k\cos{((2k-1)y)}$ where $y(0)=\dfrac{\pi}{2}$ . Determine the value of $\dfrac{y(2017)}{\pi}$ .

This problem is part of the set Extraordinary Differential Equations .

\frac{1}{3}

\frac{1}{2017}

\frac{1}{\pi}

\frac{1}{\sqrt{2}}

\frac{1}{4}

\frac{1}{2}

1 solution

Wee Xian Bin
Feb 26, 2017

Notice that if there is an equilibrium solution to the ODE, $y(x)=\frac{\pi}{2}$ for all $x$ , whereby $y'(x)=\frac{d}{dx} \frac{\pi}{2}=0$ for all $x$ , $y'(x)=\sum_{k=1}^{2017} k\cos{((2k-1)(\frac{\pi}{2}))=0}$ for all $x$ , and this also satisfies the initial condition $y(0)=\frac{\pi}{2}$ . Thus, $\frac{y(2017)}{\pi}=\frac{\pi}{2}\times\frac{1}{\pi}=\frac{1}{2}$ .

How do you know there is no other solution ?

Kushal Bose - 4 years, 3 months ago

Extraordinary Differential Equations #12

This problem is part of the set Extraordinary Differential Equations .

1 solution

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