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Homework Statement 81x^{2}y'' + 27xy' + (9x^{\frac{2}{3}}+8)y = 0 Hint: y = x1/3u x1/3 = z 2. The attempt at a solution Change of variables gives: \frac{d^{2}y}{dx^{2}} = x^{\frac{1}{3}}\frac{d^{2}u}{dx^{2}}+\frac{2}{3}x^{-\frac{2}{3}}\frac{du}{dx} - \frac{2}{9}x^{-\frac{5}{3}}u...

If I use z2 = 4x4 I get: z = 2x2 and dz/dx = z' = 4x (\frac{dy}{dx}) = \frac{dy}{dz} \frac{dz}{dx} = 4x \frac{dy}{dz} Then \frac{d}{dx}(4x\frac{dy}{dz})=4\frac{dy}{dz}+\frac{d}{dz}(\frac{dy}{dz})\frac{dz}{dx}=4\frac{dy}{dz}+4x\frac{d^{2}y}{dz^{2}} Plugging in: x2(4y' + 4xy'') +...

1. Homework Statement , relevant equation x^{2}y'' + xy' + (4x^{4}-\frac{1}{4})y = 0 2. The attempt at a solution I tried substituting z = x2 From this I have \frac{dy}{dx} = 2x \frac{dy}{dz} and \frac{d}{dx}(2x\frac{dy}{dz}) = 2xy'' + 2y' Then the original equation becomes...