(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

u''(x) - (x^6 + 3*x^2)*u(x) = 0, u(-1) = u(1) = 1, -1 <= x <= 1

3. The attempt at a solution

Consider v''(x) - s^2 * v(x) = 0, the auxiliary equation is: r^2 - s^2 = 0. Since (x^6 + 3*x^2) >= 0, s >= 0. Thus, I got r = s or -s. Suppose v(x) = a*exp(s*x)+b*exp(-s*x), then v'(x)=a*s*exp(s*x)-b*s*exp(-s*x), v''(x)=a*s^2*exp(s*x)+b*s^2*exp(-s*x)=s^2*v(x).

Let s = s(x), u''(x) - s(x)^2 * u(x) = 0, similarly I got u(x) = a*exp(s(x)*x)+b*exp(-s(x)*x). s(x) = Sqrt(x^6 + 3*x^2), u'(x) = a*(Sqrt(x^6 + 3*x^2) + x*(6*x^5+6*x)/(2*Sqrt(x^6 + 3*x^2))) * exp(s(x)*x) + b * (-Sqrt(x^6 + 3*x^2) - x * (6*x^5 + 6*x)/(2*Sqrt(x^6 + 3*x^2))) * exp(-s(x)*x), absolutely, u''(x) will be different from (x^6 + 3*x^2)*u(x), that is to say the method I used is totally incorrect!

Could anybody tell me what's wrong with my solution and how should I solve this differential equation?

Thanks a lot!

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# Homework Help: How to solve this differential equation analytically

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