The solution of the differential equation d y d x =e^ x+y +x^2 e^y is

Step-by-step solution

The given differential equation is dy dx = e^ x+y + x^2 e^y . Factoring out e^y from the right side, we get dy dx = e^y(e^x + x^2) . Separating the variables, we have dy e^y = (e^x + x^2) dx , which can be written as e^ -y dy = (e^x + x^2) dx . Integrating both sides, we get e^ -y dy = (e^x + x^2) dx . Performing the integration, we obtain -e^ -y = e^x + x^3 3 + c . Rearranging the terms, we get e^x + e^ -y + x^3 3 = -c . Since -c is an arbitrary constant, we can write this as e^x + e^ -y + x^3 3 = C . Answer: e^x+