Show that there exist a unique C^2-function y(x) defined in some region of 0 such that y(0) = 0 and sin(y(x)) + (x^2)*(e^(y(x))) = 0
What is y'(0) and y''(0)?
I know how to show that there exist a C^1 function y(x) for a function f(x,y(x)) the partial derivative of f(x,y(x)) can't be zero.
But I have actually no idea how to generalize this to check if there exist a C^2 function that satisfies the conditions. My textbook provides no example of this.
The Attempt at a Solution
I calculated the derivative of f(x,y(x)) with respect to y(x) and got:
cos(y(x))*y'(x) + (x^2)*y'(x)*e^(y(x)) = 0
After this, im not sure how to proceed.
Thanks in advance for your help!