Analytical solution of a partial system of differential equations

Question

Consider the following system of PDEs: $$\left\{ \matrix{ {{\partial f} \over {\partial y}} + {{\partial g} \over {\partial z}} = - \left( {8x + 5z} \right) \hfill \cr {{\partial f} \over {\partial x}} + {{\partial h} \over {\partial z}} = - \left( {4x + 8z} \right) \hfill \cr {{\partial g} \over {\partial x}} + {{\partial h} \over {\partial y}} = - 11xy \hfill \cr} \right.$$ We know that the analytical solution is: $$\eqalign{ & f(x,{\rm{ }}y,{\rm{ }}z){\rm{ }} = {c_1}(y,{\rm{ }}z) + {c_2}(x,{\rm{ }}z) - 4xy \cr & g(x,{\rm{ }}y,{\rm{ }}z){\rm{ }} = {c_3}(x,{\rm{ }}y) - 4xz + {\rm{ }}\int {\left( { - 5z - {{\partial {c_1}(y,{\rm{ }}z)} \over {\partial y}}} \right)dz} \cr & h(x,{\rm{ }}y,{\rm{ }}z){\rm{ }} = {c_4}\left( x \right) + (4y - 4x)z + \int {\left( { - 11xy - {{\partial {c_3}(x,y)} \over {\partial x}}} \right)dy} + {\rm{ }}\int {\left( { - {{\partial {c_2}(x,{\rm{ }}z)} \over {\partial x}} - 8z} \right)dz} \cr} $$ Because it satisfies the system. any idea how this solution could be achieved?

Answer 1

Just found a way to obtain another solution. we take the first two equation as follows: $$\left\{ \matrix{ {{\partial f} \over {\partial y}} + {{\partial g} \over {\partial z}} = - \left( {8x + 5z} \right) \to {{{\partial ^2}f} \over {\partial x\partial y}} + {{{\partial ^2}g} \over {\partial x\partial z}} = - 8 \hfill \cr {{\partial f} \over {\partial x}} + {{\partial h} \over {\partial z}} = - \left( {4x + 8z} \right) \to {{{\partial ^2}f} \over {\partial x\partial y}} + {{{\partial ^2}h} \over {\partial y\partial z}} = 0 \hfill \cr} \right.{\rm{ }}\left( 1 \right)$$

By eliminating the mutual term in Eq.1, we obtain: $$ - {{{\partial ^2}h} \over {\partial y\partial z}} + {{{\partial ^2}g} \over {\partial x\partial z}} = - 8 \to {\partial \over {\partial z}}\left( {{{\partial h} \over {\partial y}} - {{\partial g} \over {\partial x}}} \right) = 8 \to {{\partial h} \over {\partial y}} = {{\partial g} \over {\partial x}} + 8z + {c_1}\left( {x,y} \right){\rm{ }}\left( 2 \right)$$ By inserting Eq.2 in third equation of system, we have: $$\eqalign{ & {{\partial g} \over {\partial x}} + {{\partial h} \over {\partial y}} = - 11xy \to {{\partial g} \over {\partial x}} + \left( {{{\partial g} \over {\partial x}} + 8z + {c_1}\left( {x,y} \right)} \right) = - 11xy \cr & {{\partial g} \over {\partial x}} = - {{11} \over 2}xy - 4z - {{{c_1}\left( {x,y} \right)} \over 2} \to g = - {{11} \over 4}{x^2}y - 4xz - {1 \over 2}\int {{c_1}\left( {x,y} \right)dx + {c_2}\left( {y,z} \right)} \cr} $$ Now, Having g in hand, other unknowns can be obtained in a similar way.