The steps involve finding the potential function F(x,y) by integrating M with respect to x and N with respect to y, then combining the results to form the implicit solution F(x,y)=C.
Conditions: The differential equation has been verified as exact.
The steps involve finding the potential function F(x,y) by integrating M with respect to x and N with respect to y, then combining the results to form the implicit solution F(x,y)=C.
Conditions: The differential equation has been verified as exact.
When integrating a multivariable function with respect to one variable, the other variables are treated as constants. Therefore, the 'constant' of integration can be any function that depends only on the other variables, which in this case is y.
Conditions: The integration is with respect to x.; The function M depends on both x and y.
When integrating a multivariable function with respect to one variable, the other variables are treated as constants. Therefore, the 'constant' of integration can be any function that depends only on the other variables, which in this case is y.
Conditions: The integration is with respect to x.; The function M depends on both x and y.
To check if a differential equation is exact, you must verify that the partial derivative of the coefficient of dx with respect to y is equal to the partial derivative of the coefficient of dy with respect to x. If these mixed partial derivatives are equal, the equation is exact.
Conditions: The equation must be in the form M(x,y)dx+N(x,y)dy=0.; M and N must have continuous first-order partial derivatives.
To check if a differential equation is exact, you must verify that the partial derivative of the coefficient of dx with respect to y is equal to the partial derivative of the coefficient of dy with respect to x. If these mixed partial derivatives are equal, the equation is exact.
Conditions: The equation must be in the form M(x,y)dx+N(x,y)dy=0.; M and N must have continuous first-order partial derivatives.