Math Problem Solver

\[y{\left(x \right)} = \left(1 + \frac{x^{2}}{2} + \frac{x^{4}}{8}\right) C_{2} + \left(1 + \frac{x^{2}}{3}\right) C_{1} x + O\left(x^{6}\right)\]

\(C_{1}\), \(C_{2}\) are arbitrary constants.

\(O\) is the error term of the series expansion in big O notation. The error term describes the maximum size of the error near \(x = 0\) between the exact solution \(y{\left(x \right)}\), and the approximate solution \(C_{1} x \left(\frac{x^{2}}{3} + 1\right) + C_{2} \left(\frac{x^{4}}{8} + \frac{x^{2}}{2} + 1\right)\). \(y{\left(x \right)} = C_{2} \left(\frac{x^{4}}{8} + \frac{x^{2}}{2} + 1\right) + C_{1} x \left(\frac{x^{2}}{3} + 1\right) + O\left(x^{6}\right)\) means that there exists a positive constant \(M\) such that for \(x\) sufficiently close to zero \(|y{\left(x \right)} - \left(\left(1 + \frac{x^{2}}{2} + \frac{x^{4}}{8}\right) C_{2} + \left(1 + \frac{x^{2}}{3}\right) C_{1} x\right)| \le M |x^{6}|\).