BEFORE MOVING ON, WE FIRST DEFINE an n -th order ordinary equation. It is an equation for an unknown function y(x) a relationship between the unknown function and its first n derivatives.
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
This page discusses separable differential equations, emphasizing identification, variable separation, and the integration process for solutions. It addresses implicit solutions, their importance, cha...This page discusses separable differential equations, emphasizing identification, variable separation, and the integration process for solutions. It addresses implicit solutions, their importance, challenges in deriving explicit forms, and includes examples demonstrating the method.
We integrate the right-side from the initial condition x=0 to x and the left-side from the initial condition y(0)=2 to y. The first solution with x>0 of the equation \(\sin 2...We integrate the right-side from the initial condition x=0 to x and the left-side from the initial condition y(0)=2 to y. The first solution with x>0 of the equation sin2x=−1/4 places 2x in the interval (π,3π/2), so to invert this equation using the arcsine we need to apply the identity sin(π−x)=sinx, and rewrite sin2x=−1/4 as sin(π−2x)=−1/4.
