|
Chapter 5: Applications of Integration
Section 5.6: Differential Equations
|
Example 5.6.2
|
|
|
Obtain the general solution to the differential equation .
|
|
|
|
|
Solution
|
|
|
|
Mathematical Solution
|
|
Separating variables in the given differential equation leads to
|
|
|
|
|
|
|
|
|
|
|
|
|
|
The integration on the right is carried out by making the substitution so that . Thus, and the integral becomes .
Note that the solution is given in implicit form. In explicit form, there are two branches, namely, . In the shaded region in Figure 5.6.2(a), . For , the domain for each branch is the set of real numbers. For , the domain for each branch consists of . In Figure 5.6.2(b), the green curves correspond to solutions for which ; the red, to solutions for which .
|
|
|
Figure 5.6.2(a) Feasible region:
|
|
|
|
|
>
|
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: (green); (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 .
|
|
•
|
Context Panel: Solve DE
|
|
|
|
|
|
The Context Panel call to the differential equation solver results in an explicit solution for . The underlying dsolve command takes the option implicit, in which case the implicit form of the solution is returned.
|
⇒
|
|
|
|
Stepwise solution via the Student ODEs package
|
|
•
|
Tools≻Load Package: Student ODEs
|
|
Loading Student:-ODEs
|
|
•
|
Context Panel: Student ODEs≻Solve≻Generic≻General Solution with Steps≻
(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
|
|
|
|
|
|
|
|
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.
|
|
|
|
|
|
|
<< Previous Example Section 5.6
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
|
