# SECTION 4.3# Example 1de := diff(y(t),t,t) + 2*diff(y(t),t) + 2*y(t) = 0;

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P := r^2 + 2*r + 2 = 0;

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rsol := solve(P,r);

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alpha := Re(rsol[1]);

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beta := Im(rsol[1]);

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y1 := exp(alpha*t)*(cos(beta*t) + I*sin(beta*t));

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y2 := exp(alpha*t)*(cos(beta*t) - I*sin(beta*t));

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ysol := c1*y1 + c2*y2;

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subs(t=0,ysol);

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eq1 := % = 0;

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diff(ysol,t);

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subs(t=0,%);

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eq2 := % = 2;

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solve({eq1,eq2},{c1,c2});

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subs(%,ysol);

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simplify(%);

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# Solution using dsolve:de;

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dsolve(de);

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dsolve({de,y(0)=0,D(y)(0)=2});

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# Example 2de := diff(y(t),t,t) + 2*diff(y(t),t) + 4*y(t) = 0;

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P := r^2 + 2*r + 4 = 0;

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rsol := solve(P,r);

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alpha := Re(rsol[1]);

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beta := Im(rsol[1]);

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ysol := c1*exp(alpha*t)*cos(beta*t) + c2*exp(alpha*t)*sin(beta*t);

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# Solution using dsolve:de;

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dsolve(de);

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# Example 3