Lecture
A first-order differential equation has the form F ( x, y, y ' ) or y' = f ( x, y ) , where y is an unknown function of the variable x .
The function y = φ ( x ) is called the solution of a differential equation if, by substituting y = φ ( x ) and its derivative into this equation, an identity is obtained.
The set of all solutions of a differential equation is called a general solution. It is represented as some function y = φ ( x, c ) ( c is a constant). With proper selection of the constant c, the function φ ( x, c ) defines any particular solution.
The task of finding a solution to a differential equation that satisfies the initial condition y ( x 0 ) = y 0 is called the Cauchy problem.
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Mathematical analysis. Differential equations
Terms: Mathematical analysis. Differential equations