Differential Equations Study Questions and Answers

Differential Equations Study Questions and Answers What is a DE? - Answer-An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables. How are DEs classified? - Answer-By type, order, and linearity. What is the form of a "homogeneous" DE? - Answer-Mdx+Ndy = 0 is the form of an [ans] DE. The key is the right side of the equal sign is 0. What is the lecture defn. of a "homogenous" DE? - Answer-The first order DE Mdx+Ndy=0 is said to be [ans] if, when written in the form dy/dx=f(x,y) there exists an F such that f(x,y) can be written in the form F(x/y). What is the text defn. of a "homogenous" DE? - Answer-A function of x and y is [ans] if ƒ(tx,ty) = (tα)ƒ(x,y). What is an implicit DE? - Answer-An equation where the independent variable is not explicitly stated. y' = f(y) What is a "partial derivative"? - Answer-Take the derivative of a multivariable function f(x,y) with respect to one variable while treating the other variable as a constant. What is an Exact DE? - Answer-y' = f(x,y) is exact if it can be written as M(x,y) dx + N(x,y) dy = 0 where ∂M/∂y = ∂N/∂x What is the method for solving exact DEs? - Answer-1. Test for exactness 2. Integrate ∂f/∂x with respect to x, which gives a constant with respect to x, g(y) 3. Differentiate (2) with respect to y 4. Set equal to ∂f/∂y, solve for g'(y) 5. Integrate g'(y) with respect to y, giving g(y) + c