Question #40677

Let a differential equation be given by $\alpha \left( \frac{d^4y}{dx^4} \right)^5 + \beta \left( \frac{d^3y}{dx^3} \right)^7 + \gamma \left( \frac{d^2y}{dx^2} \right)^4 + \left( \frac{dy}{dx} \right)^6 = e^x$ where the coefficients are defined as: $\alpha = \sum_{n=1}^{\infty} \frac{1}{n(n+1)(n+2)} - \frac{1}{4}$ $\beta = \sum_{n=1}^{\infty} \frac{2^{n-1}}{3^n} - 1$ $\gamma = \sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ If $O$ and $D$ represent the order and degree of the differential equation respectively, find the value of $O + D$.