# SECTION 2.3# Example 1de := 1/x*diff(y(x),x) - 2*y(x)/x^2 = x*cos(x);

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de1 := de/(1/x);

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P := -2/x;

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int(P,x);

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mu := exp(%);

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de2 := mu*de1;

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#Check that the LHS is the derivative of y(x)*mu:diff(y(x)*mu,x);

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RHS := rhs(de2);

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int(RHS,x);

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sol := y(x)*mu = % + C;

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solve(sol,y(x));

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# Alternative solution using dsolve:dsolve(de);

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# Example 2de1 := diff(y(t),t) + 2*y(t) = 50*exp(-10*t);

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P := 2;

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int(P,t);

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mu := exp(%);

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de2 := mu*de;

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#Check that the LHS is the derivative of y(t)*mu:diff(y(t)*mu,t);

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RHS := rhs(de2);

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RHS := simplify(%);

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int(RHS,t);

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sol := y(t)*mu = % + C;

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solve(sol,y(t));

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simplify(%);

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sol := y = %;

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subs({t=0,y=40},sol);

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solve(%,C);

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subs(C=%,sol);

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# Alternative solution using dsolve: de1;

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dsolve(de1);

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dsolve({de1,y(0)=40});

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# Example 3Int(exp(x)*sqrt(1+cos(x)^2),x=1..2);

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evalf(%);

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exp(2)*y2 - exp(1)*y1 = %;

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subs(y1 = 4,%);

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solve(%,y2);

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# Alternative solution using dsolve:de := diff(y(x),x) + y(x) = sqrt(1+cos(x)^2);