If the transformation x=cosθreduces the differential equation (1-x^2) (d^2 y)/(d x^2)-x (d y)/(d

step 1 So this is, as you'd expect from this chapter, a separable differential equation. So we're going

If the transformation x=cosθreduces the differential...

If the transformation $x=\cos \theta$ reduces the differential equation $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+y=0$ into $\frac{d^{2} y}{d \theta^{2}}+k y=0 ;$ then the value of $k$ is

Key Concepts

Standard Form of Second Order Linear Differential Equations

This concept refers to rewriting the equation in a canonical form, such as d²y/d?² + k y = 0, which is recognizable as a simple harmonic oscillator equation. Recognizing the standard form helps in determining key parameters such as k, and in applying known solution techniques associated with such forms.

Trigonometric Substitution

This is a specific type of change of variables that uses trigonometric functions to simplify algebraic expressions, especially those involving square roots or quadratic forms. The substitution x = cos ? leverages trigonometric identities to eliminate complicating factors and to obtain a more standard form for the differential equation.

Change of Variables in Differential Equations

This concept involves rewriting a differential equation using a new variable to simplify its structure or to reveal underlying properties. In this context, the substitution x = cos ? maps the original differential equation into a new variable ?, often leading to a form that is easier to analyze or recognize.

Chain Rule for Differentiation

When the variable in a differential equation is changed, derivatives with respect to the old variable must be converted to derivatives with respect to the new one using the chain rule. This process involves expressing derivatives such as d/dx in terms of derivatives with respect to ? to correctly transform the equation.

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