Chapter 5: Applications of Integration
Section 5.6: Differential Equations
Example 5.6.2
Obtain the general solution to the differential equation x dx−2 y 1+x2 dy=0.
Solution
Mathematical Solution
Separating variables in the given differential equation leads to
2 y dy
=x dx1+x2
∫2 y ⅆy
= ∫x 1+x2−1/2 ⅆx
y2
=1+x21/22⋅1/2+c
=1+x2+c
The integration on the right is carried out by making the substitution u=1+x2 so that du=2 x dx. Thus, x dx=du/2 and the integral becomes 12∫u1/2 ⅆu.
Note that the solution is given in implicit form. In explicit form, there are two branches, namely, y±=±c+1+x2. In the shaded region in Figure 5.6.2(a), y2>0. For c>−1, the domain for each branch y± is the set of real numbers. For c<−1, the domain for each branch y± consists of x>c2−1. In Figure 5.6.2(b), the green curves correspond to solutions for which c>−1; the red, to solutions for which c<−1.
Figure 5.6.2(a) Feasible region: 1+x2+c>0
p1:=plots:-implicitplot(y^2=sqrt(1+x^2)-5,x=-5..5,y=-4..4,color=red,gridrefine=1): p2:=plots:-implicitplot(y^2=sqrt(1+x^2)-4,x=-5..5,y=-4..4,color=red,gridrefine=1): p3:=plots:-implicitplot(y^2=sqrt(1+x^2)-3,x=-5..5,y=-4..4,color=red,gridrefine=1): p4:=plots:-implicitplot(y^2=sqrt(1+x^2)-2,x=-5..5,y=-4..4,color=red,gridrefine=1): p5:=plots:-implicitplot(y^2=sqrt(1+x^2)+2,x=-5..5,y=-4..4,color=green): p6:=plots:-implicitplot(y^2=sqrt(1+x^2)+3,x=-5..5,y=-4..4,color=green): p7:=plots:-implicitplot(y^2=sqrt(1+x^2)+4,x=-5..5,y=-4..4,color=green): p8:=plots:-implicitplot(y^2=sqrt(1+x^2)+5,x=-5..5,y=-4..4,color=green): plots:-display(p||(1..8));
Figure 5.6.2(b) Solutions: c>0 (green); c<0 (red)
Maple Solution
Context Panel solution
Write the differential equation in terms of derivatives, not differentials. Maple's typesetting code infers the independent variable from the presence of y′.
Context Panel: Solve DE
x−2 y 1+x2⋅y′=0→solve DEyx=x2+1+_C1,yx=−x2+1+_C1
The Context Panel call to the differential equation solver results in an explicit solution for y±=±c+1+x2. The underlying dsolve command takes the option implicit, in which case the implicit form of the solution is returned.
dsolvex−2 y 1+x2⋅y′=0,yx,implicityx2−x2+1−_C1=0 ⇒
Stepwise solution via the Student ODEs package
Tools≻Load Package: Student ODEs
Loading Student:-ODEs
Control-drag the ODE.
Context Panel: Student ODEs≻Solve≻Generic≻General Solution with Steps≻yx (See Figure 5.6.2(a). Unfortunately, the "Solve" option is not visible in this figure.)
Figure 5.6.2(a) Context Panel access to annotated solution with steps
x−2 y 1+x2⋅y′=0→solve ODELet's solvex−2yxx2+1ⅆⅆxyx=0•Highest derivative means the order of the ODE is1ⅆⅆxyx•Separate variablesⅆⅆxyxyx=x2x2+1•Integrate both sides with respect tox∫ⅆⅆxyxyxⅆx=∫x2x2+1ⅆx+C1•Evaluate integralyx22=x2+12+C1•Solve foryxyx=x2+1+2C1,yx=−x2+1+2C1
Solution via ODE Analyzer Assistant
Syntax-free access to the differential equation solver can be obtained through the .
In the Context Panel for the differential equation, select "Solve DE Interactively" to launch the ODE Analyzer Assistant with the equation embedded.
Figure 5.6.2(c) shows the first pane. Click the button "Solve Symbolically" to move to the second pane, an image of which is given in Figure 5.6.2(d).
Figure 5.6.2(c) Opening pane of ODE Assistant
On the left in the Solve Symbolically pane, change the Explicit setting from auto to no, as per Figure 5.6.2(d).
Click the Solve button to obtain the solution displayed in the upper right, Figure 5.6.2(d).
Figure 5.6.2(d) Second pane of ODE Assistant
Check the box for "Show Maple commands", lower left in Figure 5.6.2(d). The Maple dsolve command is displayed, and can be returned if selected as the object returned when the Quit button is pressed.
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