you cannot use dsolve, you should integrate. 1-y;

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int(1/%,y);

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As y(0)=3, 1-y is negative: we have to change the sign under the argument of the logaritm. -ln(y-1);

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%=int(x^3,x)+C;

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sol:=y(x)=solve(%,y);

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3=subs(x=0,rhs(sol));

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C=solve(%,C);

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sol:=subs(%,sol);

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simplify(%);

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restart: The equation is not in standard form. It is imperative that the linear equation be in standard form before we attempt to determine the coefficient p of y. If the coefficient of y' is not one, then p and the resulting integrating factor will be incorrect; and the "solution" will not check. It is also necessary that the term involving y be on the same side as y' and that all x's be on the right hand side. Here is the original differential equation. odiffeq:=x*diff(y(x),x)+3*y(x)+2*x^2=x^3+4*x;

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To rewrite it in standard form, we subtract 2*x^2 from both sides, then we divide by x. (In this simple case, it may be easier to retype the equation by hand.) ndiffeq:=diff(y(x),x)+(3*y(x))/x=x^2+4-2*x;

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The expression p is the coefficient of y, including sign, when the equation is in standard form. p:=3/x;

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int(p,x);

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The integrating factor is the exponential of the antiderivative of p. mu:=exp(%);

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After the standard form equation is multiplied by the integrating factor, the left hand side is a perfect derivative. Hence, when we integrate, the left hand side will always be mu*y(x). To find y(x), we need only divide by mu. Note that Maple does not automatically insert a constant of integration, so we must do it manually. mu*ndiffeq;

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expand(%);

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mu*y(x)=int(rhs(%),x)+c;

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solve(%,y(x)): sol:=y(x)=expand(%); Notice that x is different from 0 as we divided from it in one of the previous steps.

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Checking (in the original differential equation): subs(sol,odiffeq): simplify(%);

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There are two approaches to solve this kind of situations. The 1st one is to consider two different differential equations. We use the 2nd one as dsolve command accepts the piecewise entry, producing a piecewise answer. restart: P:=piecewise(0<=x and x<=2,1,x>2,3);

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diffeq:=diff(y(x),x)+P*y(x)=x;

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inits:=y(0)=1;

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sol:=dsolve({diffeq,inits},y(x)); Notice that the program is computing the backward solution as well, just by putting P(x)=0 for x<0. The backward solution is meaningless fopr us, as our differential problem lives in the halfline [0,+infty).

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plot(rhs(sol),x=0..5);

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