Learning Guide Learning Guide

6.2 Ordinary Differential Equations • 233{y(x) = e x _C2 + e (−x) _C1 − 1}To specify initial conditions for the derivative of a function, use thefollowing notation.D(fcn)(var_value) = value(D@@n)(fcn)(var_value) = valueThe D notation represents the derivative. The D@@n notation represents thenth derivative. Here is a differential equation and some initial conditionsinvolving the derivative.> de1 := diff(y(t),t$2) + 5*diff(y(t),t) + 6*y(t) = 0;de1 := ( d2dt 2 y(t)) + 5 ( d y(t)) + 6 y(t) = 0dt> ini := y(0)=0, D(y)(0)=1;ini := y(0) = 0, D(y)(0) = 1Again, use dsolve to find the solution.> dsolve( {de1, ini}, {y(t)} );y(t) = e (−2 t) (−3 t)− eAdditionally, dsolve may return a solution in parametric form,[x=f(_T), y(x)=g(_T)], where _T is the parameter.The explicit Option Maple may return the solution to a differentialequation in implicit form.> de2 := diff(y(x),x$2) = (ln(y(x))+1)*diff(y(x),x);de2 := d2dy(x) = (ln(y(x)) + 1) (dx2 dx y(x))> dsolve( {de2}, {y(x)} );