Chapter 5: Applications of Integration
Section 5.6: Differential Equations
Solve the initial-value problem consisting of the differential equation x−2 y 1+x2⋅y′=0 and the initial condition y1=2. Graph the solution.
From Example 5.6.2, the general solution of the given differential equation can be written in the implicit form y2=1+x2+c.
Applying the initial condition y1=2 to this solution results in the equation
from which it follows that c=4−2 and
Now, to obtain yx explicitly requires some care.
Taking the square root leads to
Figure 5.6.3(a) Solution of the give IVP
However, since y1=2>0, the solution of this IVP must be y=1+x2+4−2.
This solution is graphed in Figure 5.6.3(a).
The implicit form of the solution can also be obtained by separating variables in the differential equation, and then evaluating the definite integrals detailed in the remarks after Table 5.6.1.
∫2y2 s ⅆs
Of course, the explicit solution yx that satisfies the initial condition is obtained from the implicit solution with the strategy used above. Note that it would be misleading to declare the implicit form as the solution of the IVP because the implicit form contains both branches, including the branch that does not satisfy the initial condition, and is therefore not a solution of the IVP.
Solution via Context Panel
Control-drag the differential equation.
Press the Enter key.
Add an Initial Condition≻y1=2
(See Figure 5.6.3(b).)
Context Panel: Solve DE≻yx
Context Panel: Right-hand Side
Context Panel: Plot Builder≻2-D plot
2-D Options: scaling≻constrained
Figure 5.6.3(b) "Add Condition" dialog
x−2 y 1+x2⋅y′=0
→add initial condition
→right hand side
Stepwise solution via the Student ODEs package
Tools≻Load Package: Student ODEs
Control-drag the ODE.
Form a set with the ODE and the initial condition.
Apply the ODESteps command.
(This option is not available in the Context Panel.)
ODEStepsx−2 y 1+x2⋅y′=0,y1=2
Let's solvex−2⁢y⁡x⁢x2+1⁢ⅆⅆxy⁡x=0,y⁡1=2•Highest derivative means the order of the ODE is1ⅆⅆxy⁡x•Separate variablesⅆⅆxy⁡x⁢y⁡x=x2⁢x2+1•Integrate both sides with respect tox∫ⅆⅆxy⁡x⁢y⁡xⅆx=∫x2⁢x2+1ⅆx+C1•Evaluate integraly⁡x22=x2+12+C1•Solve fory⁡xy⁡x=x2+1+2⁢C1,y⁡x=−x2+1+2⁢C1•Use initial conditiony⁡1=22=2+2⁢C1•Solve for_C1C1=−22+2•Substitute_C1=−22+2into general solution and simplifyy⁡x=x2+1−2+4•Use initial conditiony⁡1=22=−2+2⁢C1•Solution does not satisfy initial condition•Solution to the IVPy⁡x=x2+1−2+4
Solution via the ODE Analyzer Assistant
Control-drag the differential equation.
Context Panel: Solve DE Interactively
x−2 y 1+x2⋅y′=0→solve DE interactively
The first pane of the
assistant is shown in Figure 5.6.8. The Context Panel automatically inserts the differential equation into the Assistant.
To add the initial condition, click the Edit button at the bottom of the "Conditions" window.
Figure 5.6.3(d) shows the Edit Conditions pane. Fill in the data under "Add Condition", then click both the Add and the Done buttons.
Press the "Solve Symbolically" button to obtain the "Solve Symbolically" pane shown in Figure 5.6.3(f). Click the Solve button to obtain the analytic solution of the IVP.
Figure 5.6.3(c) Pane 1, ODE Analyzer Assistant
Figure 5.6.3(d) Edit-Conditions pane
Figure 5.6.3(e) shows the Plot Options pane, launched by clicking the Plot Options button in the Solve Symbolically pane. Change the default values for the range of the independent variable.
Press the Plot button in the Solve Symbolically pane to obtain a graph of the solution.
Figure 5.6.3(e) Plot Options pane
Figure 5.6.3(f) Solve Symbolically pane
Note the "Show Maple commands" check-box with which the dsolve and plot commands used by the assistant can be seen. The "On Quit, Return" drop-down can be used to select what will be returned to the worksheet when the Quit button is pressed. The graph, the analytic solution, the Maple commands, or nothing, can be thereby returned.
Solution via the dsolve command
dsolvex−2 y 1+x2⋅y′=0,y1=2,yx
To graph the solution returned by the dsolve command, just the right-hand side of the equation yx=…is given to the plot command or to the Context Panel's Plots option. For this, the rhs (right-hand side) command, or the Context Panel's "Right-hand Side" option can be used.
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