Here's the solution to Section 2.3, Problem 49.

We want to express the solution to x^3 y' + 2x^2 y = 10 sin(x), y(1)=0

in terms of the sine integral (See p 66-67).

Maple will solve this directly, but let's try to do it ourselves... We'll write

the express in standard form,

y' + (2/x) y = 10 sin(x) / x^3

Then the integral of p(x) is: 2 ln( x ), and the integrating factor is x^2.

Therefore, to solve this ODE, we write:

(x^2 y ) ' = 10 sin(x) / x

x^2 y = 10 int( sin(x)/x) dx

y = (10/x^2)* Si(x) + C/x^2

Let's see what Maple says:

DE1:=diff(y(x),x)+(2/x)*y(x)=10*sin(x)/x^3;

Y:=dsolve(DE1,y(x));

To see what the Si(x) function is, use Maple's help file:

?Si

We'll plot the solution for x>0. First, solve for the arbitrary constant:

F:=rhs(Y);

F1:=subs(x=1,F);

solve(F1=0,_C1);

Our function is (I'll just re-type it with the found value of the constant):

G:=(10/x^2)*( Si(x)-Si(1) );

plot(G,x=0.1..5,y=-20..5);

To find the maximum of this function, set the derivative = zero and solve for x:

Eqn1:=diff(G,x)=0;

solve(Eqn1,x);

To get a numerical solution, use fsolve and give Maple an interval (guess from our previous graph)

fsolve(Eqn1,x=1..2);