Differential equation

[BenBen]

In problem d) the differential equation (*) can be solved with an exact substitution of y(x)=x^(-1/2)*u(x). We´re gonna show that u(x) satisfies the equation u``+u=0.

My problem is that I don´t get the question. Can someone please help me get started?

 

[Romsek]

Just plug [MATH]y(x) = x^{-1/2}u(x)[/MATH] into (*)

you'll need to find [MATH]y'(x)[/MATH] and [MATH]y''(x)[/MATH] and plug them in.

Things should simplify until you end up with

[MATH]x^{3/2}(u''(x) + u(x)) = 0[/MATH]
and thus [MATH]u''(x) + u(x)=0[/MATH]