HardMathematicsHigh School (11–12)Functions and Calculus › Continuity and DifferentiabilityNumerical

Let y = e^(2 sin⁻¹x) for -1 < x < 1. Show that (1 - x²)y'' - xy' - 4y = 0…

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

Let $y = e^{2\sin^{-1}x}$ for $-1 < x < 1$ . Show that $(1 - x^2)y'' - xy' - 4y = 0$ . Then use the relation, at $x = 0$ , to find $y(0)$ , $y'(0)$ and $y''(0)$ . This avoids a second long differentiation. Give each as a number.

Need a reminder? The key ideaShow

$f''$ is the derivative of $f'$ , and $f^{(n)}$ is the $n$ th derivative. For a parametric curve, $\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)\Big/\dfrac{dx}{dt}$ . To verify a relation, differentiate twice and substitute.

The worked solution

Have you tried it?

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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