Chapter 5: Applications of Integration
Section 5.6: Differential Equations
Obtain the general solution of the differential equation y′x=x yx.
The differential equation is variable-separable; it can be written in the form y′x/yx=x so that fy=1/y and gx=x. In this form, the equation yields immediately to antidifferentiation:
= ∫x ⅆx
= C ex2/2
To go from the second to the third equality, exponentiate both sides: elny=y by the definition of the logarithm. On the right, be sure to exponentiate the sum. A common error is to write this as the sum of exponentials.
Note the use of the law of exponents: ea+b=ea⋅eb. Finally, since c is arbitrary, so is ec. It is generally wise to give a new name to a "new" arbitrary constant.
Finally, note that an arbitrary constant is added to just one side of the equation that results from integrating both sides of the separated differential equation. If arbitrary constants were added to both sides, the one on the left, say, could be subtracted from both sides, and on the right, the difference of two arbitrary constants would still be just one arbitrary constant.
Context Panel solution
Control-drag the differential equation
Context Panel: Solve DE Interactively
y′x=x yx→solve DEy⁡x=_C1⁢ⅇ12⁢x2
Maple uses _C1 for the arbitrary constant. Names generated internally by Maple usually start with an underscore. For this reason, users are cautioned not to form names with a leading underscore.
Stepwise solution via the Student ODEs package
Tools≻Load Package: Student ODEs
Control-drag the ODE.
Context Panel: Student ODEs≻Solve≻Generic≻General Solution with Steps≻yx
(See Figure 5.6.1(a). Unfortunately, the "Solve" option is not visible in this figure.)
Figure 5.6.1(a) Context Panel access to annotated solution with steps
y′x=x yx→solve ODELet's solveⅆⅆxy⁡x=x⁢y⁡x•Highest derivative means the order of the ODE is1ⅆⅆxy⁡x•Separate variablesⅆⅆxy⁡xy⁡x=x•Integrate both sides with respect tox∫ⅆⅆxy⁡xy⁡xⅆx=∫xⅆx+C1•Evaluate integralln⁡y⁡x=x22+C1•Solve fory⁡xy⁡x=ⅇx22+C1
Below, a stepwise Maple solution is implemented with commands, then interactively via the Context Panel system.
A stepwise Maple solution implemented with commands
Assign the differential equation to the name q1.
Divide by yx.
(The variables are now separated.)
Obtain the antiderivative of both sides.
(The map command places the int command onto each side of the equation.)
Add a constant of integration to the right side.
(The "equation" 0=c is added to the equation q3 as a device to append c to the right-hand side.)
Exponentiate both sides.
(The map command applies the exponential function exp to both sides of equation q4.)
Apply the appropriate law of exponents.
(The expand command converts ea+b to ea⋅eb.)
Replace ec with C.
(Evaluate equation q6 with ec set equal to C.)
Interactive stepwise solution via the Context Panel system
Control-drag (or copy/paste) the differential equation.
Press the Enter key.
Use the equation label to divide by yx.
Press the Enter key.
Context Panel: Map Command Onto
Figure 5.6.1(a) Dialog for Map Command Onto
To equation , add the "equation" 0=c and press the Enter key.
Solve≻Isolate Expression for≻yx
Context Panel: Expand≻Expand
→isolate for y(x)
The traditional approach to solving separable differential equations makes liberal use of the differentials dy and dx. Maple does not use this notation for differentials, except within typeset integrals where the differentials are written by Maple, and not by the user. Hence, the stepwise approach to solving a separable differential equation necessarily deviates from the standard textbook approach.
Notice how the integral on the left, that is, the integral of y′/y, is in terms of the independent variable x. That's because the differential equation itself carries the independent variable explicitly. When the variables are separated, Maple still sees yx, not just the symbol y. To integrate dy/y on the left would require a rewrite of the left side to just 1/y and then it would be impossible to map integration across both sides of the equation - the left side is integrated with respect to y, but the right side is integrated with respect to x.
Hence, rigorously following the textbook methodology is a lot more work. If Maple is to be used to implement the solution stepwise, accommodations must be made for the structure and functioning of Maple's tools.
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