# Initial Condition

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## initial condition

[i′nish·əl kən′dish·ən]## Initial Condition

in the mathematical analysis of a process the state of the process at a given moment of time, taken as the initial moment. If the process is described by a differential equation, then the problem of finding solutions for a given initial condition is called the Cauchy problem. An initial condition for the equation

consists in specifying *y, dy/dt*, . . . , (*d*^{n-1}*y)/dt*^{n-1} for a value *t* = *t*_{0}. If *n* = 2 and *y* = *y(t)* is the law of motion of a point mass, then the initial condition specifies the position and velocity of the point at the moment *t* = *t*_{0} An initial condition for a partial differential equation is similarly defined. Thus, for the equation

of a free vibrating string, where *u(t, x)* is the deviation of the point *x* of the string at the moment *t* from the position of equilibrium on the *x*-axis, the initial condition specifies the initial shape *u*ǀ*t* = *t*_{0} = *f(x*) of the string and the initial velocities δ*u/* δ/1,= *t*_{0} of the points of the string. Any other argument may play the role of time. An initial condition is then specified for some value of that argument.