# SECTION 4.4# Example 1de := diff(y(t),t,t) + 3*diff(y(t),t) + 2*y(t) = 3*t;

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y2 := A*t+B;

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subs(y(t) = y2,de);

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eq := expand(%);

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eq1 := 2*A = 3;

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eq2 := 3*A + 2*B = 0;

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solve({eq1,eq2},{A,B});

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subs(%,y2);

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# Alternative way to get eq1 and eq2:diff(eq,t);

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subs(t=0,eq);

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# Example 2de2 := lhs(de) = 10*exp(3*t);

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yp := A*exp(3*t);

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subs(y(t) = yp,de2);

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expand(%);

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eq := %/exp(3*t);

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simplify(%);

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solve(%,A);

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subs(A=%,yp);

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# Example 3de3 := lhs(de) = sin(t);;

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yp := A*sin(t) + B*cos(t);

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subs(y(t) = yp,de3);

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expand(%);

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eq1 := A - 3*B = 1;

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eq2 := B + 3*A = 0;

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solve({eq1,eq2},{A,B});

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subs(%,yp);

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# Example 4de4 := diff(y(t),t,t) + 4*y(t) = 5*t^2*exp(t);

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yp := (A*t^2+B*t+C)*exp(t);

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subs(y(t) = yp,de4);

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expand(%);

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%/exp(t);

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simplify(%);

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eq := %;

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eq1 := subs(t=0,eq);

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diff(eq,t);

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eq2 := subs(t=0,%);

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diff(eq,t,t);

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eq3 := %/2;

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solve({eq1,eq2,eq3},{A,B,C});

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subs(%,yp);

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# Example 5de5 := diff(y(t),t,t) + 2*diff(y(t),t) - 3*y(t) = f;

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fa := 7*cos(3*t);

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de5a := subs(f=fa,de5);

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dsolve(de5a);

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fb := 2*t*exp(t)*sin(t);

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de5b := subs(f=fb,de5);

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dsolve(de5b);

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fc := t^2*cos(pi*t);

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de5c := subs(f=fc,de5);

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dsolve(de5c);

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fd := 5*exp(-3*t);

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de5d := subs(f=fd,de5);

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dsolve(de5d);

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fe := 3*t*exp(t);

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de5e := subs(f=fe,de5);

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dsolve(de5e);

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ff := t^2*exp(t);

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de5f := subs(f=ff,de5);

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dsolve(de5f);

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# Example 7de7 := diff(y(t),t,t) - 2*diff(y(t),t) + 2*y(t) = 5*t*exp(t)*cos(t);

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dsolve(de7);

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