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Solution to "5.2", 2.c.

> with(DEtools):Order:=12;

Order := 12

> dsolve({diff(y(x),x$2)+x*diff(y(x),x)-y(x)=0,y(0)=2,D(y)(0)=1},y(x),series);

y(x) = (series(2+x+x^2-1/12*x^4+1/120*x^6-1/1344*x^8+1/17280*x^10+O(x^12),x,12))

> y3:= x -> 2+x+x^2;

y3 := proc (x) options operator, arrow; 2+x+x^2 end proc

> y5:= x -> 2+x+x^2-1/12*x^4;

y5 := proc (x) options operator, arrow; 2+x+x^2-1/12*x^4 end proc

> y7:= x -> 2+x+x^2-1/12*x^4+1/120*x^6;

y7 := proc (x) options operator, arrow; 2+x+x^2-1/12*x^4+1/120*x^6 end proc

> y9:= 2+x+x^2-1/12*x^4+1/120*x^6-1/1344*x^8;

y9 := 2+x+x^2-1/12*x^4+1/120*x^6-1/1344*x^8

> plot([y3(x),y5(x),y7(x),y9(x)],x=0..5,color=[red,green,yellow,blue],thickness=3);

[Plot]

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