30a

>	dsolve(diff(y(x),x$2)+y(x)=0);

>	f:=(Ax+B)*exp(-x);

>	diff(f,x$2)+f;

>	f:=x*(A*cos(x)+B*sin(x));

>	ev:=diff(f,x$2)+f;

>	simplify(ev);

>

30b

>	f:=exp(x)*((A*x^5+B*x^4+C*x^3+D*x^2+E*x+F)*sin(x)+(G*x^5+H*x^4+J*x^3+K*x^2+L*x+M)*cos(x));

>	ev:=diff(f,x$2)+f;

>	col:=collect(expand(ev),{sin(x),cos(x),exp(x)});

>	p1:=collect([op([op(col)][1])][1],x);

>	p2:=collect([op([op(col)][2])][1],x);

31

>	de:=L*C*diff(v(t),t$2)+R*C*diff(v(t),t)+v(t)=cos(omega*t);

>	dsolve(de);

32

>	hom:=diff(y(x),x$2)+4*y(x)=0;

>	dsolve(hom);

>	y1:=sin(2*x);

>	y2:=cos(2*x);

>	g:=4*tan(2*x);

>	y1p:=diff(y1,x);

>	y2p:=diff(y2,x);

>	u1p:=-y2*g/(y1*y2p-y2*y1p);

>	u2p:=-y1*g/(y1*y2p-y2*y1p);

>	u1p:=simplify(u1p);

>	u2p:=simplify(u2p);

>	int(u1p,x);

>	u2p2:=2*(1/cos(2*x)-cos(2*x));

>	int(2/cos(2*x),x);

>	int(2*cos(2*x),x);