Maple Worksheet- Section 2.2 (Separable Equations)

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Now we want to solve for the x-intercepts we see in the graph-

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Sorry about the following change in font color- I think I like this better:

Y14:=dsolve({DE14,y(0)=1},y(x));

plot(rhs(Y14),x=-sqrt(5)/2..sqrt(5)/2,y=0..7);

DE16:=diff(y(x),x)=x*(x^2+1)/(4*(y(x))^3);

Y16:=dsolve({DE16,y(0)=-1/sqrt(2)},y(x));

plot(rhs(Y16),x=-6..6);

To see the full hyperbola in Problem 16, we could implicitly plot the curve:

with(plots):

implicitplot(y^4=(1/4)*(x^2+1)^2,x=-6..6,y=-5..5,grid=[50,50]);

DE20:=(y(x))^2*sqrt(1-x^2)*diff(y(x),x)=arcsin(x);

Y20:=dsolve({DE20,y(0)=1},y(x));

plot(rhs(Y20),x=-1..1);

In Problem 21, we need to determine where the solution (or integral curve) has

a vertical tangent. As in the HW solutions, this will be the solution to the following equations:

solve(x^3+x-2=0,x);

solve(x^3+x+2=0,x);

In Problem 25, solve the IVP and find the max:

DE25:=diff(y(x),x)=2*cos(2*x)/(3+2*y(x));

Y25:=dsolve({DE25,y(0)=-1},y(x));

dY25:=diff(rhs(Y25),x);

plot(rhs(Y25),x=-1..1.5);

From this plot of the solution, we see that the maximum is between 0.5 and 1:

fsolve(dY25=0,x=0.5..1);

In Problem 27, we'll go ahead and have Maple solve the DE so we don't have to enter it by hand.

We will then solve for y_0:

with(DEtools):

DE27:=diff(y(t),t)=t*y(t)*(4-y(t))/3;

dfieldplot(DE27,y(t),t=-1..7,y=-4..9);

Y27:=dsolve({DE27,y(0)=1/2},y(t));

To solve the last question in 27, we need to know t for which this solution crosses 3.98.

To get a rough estimate, we'll plot y(t)-3.98 (and look for the zero crossing):

plot(rhs(Y27)-3.98,t=-1..7);

From the picture, the solution is about 3. We'll use this estimate in Maple:

fsolve(rhs(Y27)=3.98,t=2..4);