HardMathematicsHigh School (11–12)Functions and CalculusNumerical

A designer studies how a quantity y scales with the size x of a model…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A designer studies how a quantity $y$ scales with the size $x$ of a model. She finds that, for $x > 0$ , the quantity satisfies $x^2y'' - 2xy' - 4y = 0$ . She guesses solutions of the form $y = x^m$ , where $m$ is a constant, and puts the guess into the equation. This gives an equation for $m$ with two roots. She calls the larger root $m_1$ and the smaller root $m_2$ . The general solution is $y = Ax^{m_1} + Bx^{m_2}$ . A measurement gives $y(1) = 6$ and $y'(1) = 1$ . Find $m_1$ and $m_2$ , then the constants $A$ and $B$ , and then $y(2)$ .

Need a reminder? The key ideaShow

The order is the order of the highest derivative. The degree is its power, once roots and fractions of derivatives are cleared. A general solution has as many constants as the order, and conditions fix them.

\displaystyle \text{order } n \ \Rightarrow\ n \text{ constants}.

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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