Example 4: Solve [Maple Math] .

Solution: The roots of the characteristic equation are

> solve(r*(r-1)+2*r-12=0);

[Maple Math]

Hence two independent solutions of the homogeneous equation are [Maple Math] and [Maple Math] . We find a particular solution using the method of variation. Recall that for this method to work the differential equation has to be in the standard form [Maple Math] . Thus new [Maple Math] for Euler equation is equal to [Maple Math] .

> y[1]:=x->x^(-4);y[2]:=x->x^3;

[Maple Math]

[Maple Math]

> solve({y[1](x)*dc[1]+y[2](x)*dc[2]=0,diff(y[1](x),x)*dc[1]+diff(y[2](x),x)*dc[2]=sqrt(x)/x^2},{dc[1],dc[2]});

[Maple Math]

> c[1]:=int(-1/7*x^(7/2),x);c[2]:=int(1/7/x^(7/2),x);

[Maple Math]

[Maple Math]

So the particular solution is

> c[1]*y[1](x)+c[2]*y[2](x);

[Maple Math]

Hence the general solution to the nonhomogeneous equation is [Maple Math] .

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