## Maple Tutorial: Numerical Methods for approximating solutions of Differential Equations (DEs)

The Euler approximation method compares well with the exact solution. Concludes that the maple leaf / tutorial. Continue with the tendency to calculate your toolbox Toolbox ….

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Once the condition that the solution must go through the origin (as done whe n dsolve was used above) is added to the syntax, notice the difference between the plot found using disolve and the plot of the solution in the slope field. Remember, when using slope fields or numerical methods in general, the output from these only produce approximations (so be careful).

**EULER\’S METHOD** Recall that the linearization of a **differentiable function** at a point (the linearization is the equation of the tangent line at the point) is a good approximation to the function near the point. When dyn)). Keep in mind that the method only provides approximates to the solution whose accuracy depends on dx and the number of steps used (n).

Errors may grow if n is too large (or dx is t oo big). See section 6.12 in text for a more detailed description of t he method. Here we utilize a Maple s ubrouti ! – 2 x y1 + x2 e uler_approx:=proc (f ,x_start,y_start,dx ,n_total)local x,y,Y,i;x[1]:=x_start;y[1]:=y_sta rt;for i from 1 to n_total do y[i+1 ]:=y[i]+f(x[i],y[i] )* dx; x[i+1 ]:=x[1]+i*dx;end do;Y:=[seq([x [k],y[k]],k=1..n_to tal)];return(Y

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