Differential Equations Mind Map

The order of a differential equation is the order of the highest derivative present. The degree is the power of the highest order derivative after the equation is made polynomial in derivatives, if such a polynomial form is possible.

Before writing degree, check whether the equation is polynomial in derivatives. If a derivative is inside a square root, denominator, or trigonometric function and cannot be converted to polynomial form, degree is not defined.

Forming a differential equation means eliminating arbitrary constants from a given family of curves by differentiating enough times and combining the resulting equations.

Count the arbitrary constants first. Differentiate that many times, then use the original equation and derivative equations together to remove every arbitrary constant.

A first-order differential equation is variable separable if it can be arranged so that all terms involving y and dy are on one side and all terms involving x and dx are on the other side.

After separating variables, check that one side contains only y with dy and the other contains only x with dx before integrating.

A first-order differential equation dy/dx = F(x, y) is homogeneous when the right side can be expressed as a function of y/x or x/y. It is solved by substituting y = vx or x = vy to reduce it to a variable-separable equation.

Use y = vx when the expression naturally contains y/x. Use x = vy when x/y makes the equation simpler. Always apply the product rule after substitution.

A first-order linear differential equation in y has the form dy/dx + P(x)y = Q(x), where P and Q are functions of x. It is solved using the integrating factor e^(∫P(x) dx).

First divide by the coefficient of dy/dx so that its coefficient becomes 1. Only then identify P and Q and calculate the integrating factor.