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I would like to solve the PDE

$$\partial_{x}f(x,y) + f(x,y)^2 = V(x,y)$$

with $f(0,0)=0$ and $\partial_y f(0,0)=0$ using a power series ansatz, i.e. I have an explicit expression for $V(x,y)=\sin(x+y)\cos(x)\cos(2y)$ and want to find the first coefficients of a power series $f(x,y) = \sum_{n} c_n x^{n_1}y^{n_2}$ solving this equation. I wonder if there is a way in mathematica to do this to get the first let's say $\vert n\vert\le 11$ coefficients.

I would like to solve the PDE

$$\partial_{x}f(x,y) + f(x,y)^2 = g(x,y)$$

with $f(0,0)=0$ and $\partial_y f(0,0)=0$ using a power series ansatz, i.e. I have an explicit expression for $g(x,y)=\sin(x+y)\cos(x)\cos(2y)$ and want to find the first coefficients of a power series $f(x,y) = \sum_{n} c_n x^{n_1}y^{n_2}$ solving this equation. I wonder if there is a way in mathematica to do this to get the first let's say $\vert n\vert\le 11$ coefficients.

I would like to solve the PDE

$$\partial_{x}f(x,y) + f(x,y)^2 = V(x,y)$$

with $f(0,0)=0$ and $\partial_y f(0,0)=0$ using a power series ansatz, i.e. I have an explicit expression for $V(x,y)=\sin(x+y)\cos(x)\cos(2y)$ and want to find the first coefficients of a power series $f(x,y) = \sum_{n} c_n x^{n_1}y^{n_2}$ solving this equation.

I would like to solve the PDE

$$\partial_{x}f(x,y) + f(x,y)^2 = V(x,y)$$

with $f(0,0)=0$ and $\partial_y f(0,0)=0$ using a power series ansatz, i.e. I have an explicit expression for $V(x,y)=\sin(x+y)\cos(x)\cos(2y)$ and want to find the first coefficients of a power series $f(x,y) = \sum_{n} c_n x^{n_1}y^{n_2}$ solving this equation. I wonder if there is a way in mathematica to do this to get the first let's say $\vert n\vert\le 11$ coefficients.