yleisratkaisu

In Finnish mathematical language, yleisratkaisu refers to the general solution of a differential equation. It is the complete family of all functions that satisfy the equation, typically containing one or more arbitrary constants. The term is used to distinguish this broad solution set from particular solutions, which are obtained by imposing initial or boundary conditions that determine the constants.

For ordinary differential equations, the general solution is often found by solving the associated homogeneous equation to obtain the complementary solution and then adding a particular solution that satisfies the non-homogeneous part. The sum, when constants are included, forms the general solution. Example: dy/dx = y has general solution y = C e^x.

For higher-order equations, the general solution contains as many arbitrary constants as the order of the equation. An initial condition or boundary condition reduces these constants to a unique, single solution. Example: y'' - y = 0 has general solution y = C1 e^x + C2 e^{-x}.

Sometimes differential equations admit singular solutions that are not obtained by simply varying the constants in the general form. Such solutions are distinct from the yleisratkaisu and require separate analysis.

Origin and usage: yleisratkaisu is formed from Finnish yleis (general) and ratkaisu (solution). It is standard in Finnish textbooks when discussing solving strategies for differential equations and is also encountered in Finnish-language educational materials and research contexts.