# SECTION 4.8# Example 1de := m*diff(y(t),t,t) + b*diff(y(t),t) + k*y(t) = 0;

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de1 := subs(m=1/8,b=0,k=16,de);

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omega := subs({m=1/8,k=16},sqrt(k/m));

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simplify(%);

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omega := %;

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ysol := C1*cos(omega*t) + C2*sin(omega*t);

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subs(t=0,ysol);

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eq1 := simplify(%) = 1/2;

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diff(ysol,t);

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subs(t=0,%);

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eq2 := simplify(%) = sqrt(2);

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solve({eq1,eq2},{C1,C2});

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subs(%,ysol);

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ysol := %;

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C1 := 1/2; C2 := 1/8;

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A := sqrt(C1^2 + C2^2);

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phi := arctan(C1/C2);

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phi := evalf(%);

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fsolve(ysol=0,t);

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# Solution using dsolve:dsolve(de1);

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dsolve({de1,y(0)=1/2,D(y)(0)=sqrt(2)});

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# Example 2de2_case1 := subs({m=1,b=6,k=25},de);

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P := r^2+6*r+25 = 0;

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solve(P,r);

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C1 := 'C1'; C2 := 'C2';

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ysol := exp(-3*t)*(C1*cos(4*t) + C2*sin(4*t));

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subs(t=0,ysol);

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eq1 := simplify(%) = 1;

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diff(ysol,t);

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subs(t=0,%);

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eq2 := simplify(%)=0;

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solve({eq1,eq2},{C1,C2});

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subs(%,ysol);

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ysol_case1 := %;

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C1 := 1; C2 := 3/4;

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A := sqrt(C1^2 + C2^2);

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phi := arctan(C1/C2);

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phi := evalf(%);

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de2_case2 := subs({m=1,b=10,k=25},de);

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P := r^2+10*r+25=0;

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solve(P,r);

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C1 := 'C1'; C2 := 'C2';

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ysol := (C1+C2*t)*exp(-5*t);

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subs(t=0,ysol);

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eq1 := simplify(%)=1;

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diff(ysol,t);

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subs(t=0,%);

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eq2 := simplify(%) = 0;

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solve({eq1,eq2},{C1,C2});

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subs(%,ysol);

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ysol_case2 := %;

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de2_case3 := subs({m=1,b=12,k=25},de);

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P := r^2+12*r+25 = 0;

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sol := solve(P,r);

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r1 := sol[1]; r2 := sol[2];

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C1 := 'C1'; C2 := 'C2';

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ysol := C1*exp(r1*t) + C2*exp(r2*t);

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subs(t=0,ysol);

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eq1 := % = 1;

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diff(ysol,t);

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subs(t=0,%);

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eq2 := % = 0;

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solve({eq1,eq2},{C1,C2});

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subs(%,ysol);

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ysol_case3 := %;

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# Example 3de3 := subs({m=1/4,b=1,k=4},de);

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P := 1/4*r^2 + r + 4 = 0;

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solve(P,r);

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alpha := -2; beta := 2*sqrt(3);

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ysol := exp(alpha*t)*(C1*cos(beta*t) + C2*sin(beta*t));

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subs(t=0,ysol);

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eq1 := %=-1/2;

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diff(ysol,t);

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subs(t=0,%);

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eq2 := % = -1;

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sol := solve({eq1,eq2},{C1,C2});

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subs(%,ysol);

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ysol := %;

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C1 := rhs(sol[1]); C2 := rhs(sol[2]);

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A := sqrt(C1^2+C2^2);

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phi := arctan(C1,C2);

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ysolp := diff(ysol,t);

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simplify(%);

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fsolve(%=0,t);

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subs(t=%,ysol);

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evalf(%);

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abs(%);

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