Unformatted text preview: 2 x + y2 (x2 + y 2 )2 y2 y 2 (x2 − y 2 ) x2 y 2 (x2 − y 2 ) −2 +8 x2 + y 2 (x2 + y 2 )2 (x2 + y 2 )3 240 • Chapter 7: Solving Calculus Problems At (0, 0), you must use the limit deﬁnition.
> Limit( ( fy(h, 0)  fy(0,0) )/h, h=0 ); h→0 lim 1 > fyx(0,0) := value( % ); fyx(0, 0) := 1 Note that the two mixed partial derivatives are diﬀerent at (0, 0).
> fxy(0,0) <> fyx(0,0); −1 = 1 The mixed partial derivatives are equal only if they are continuous. As you can see in the plot of fxy, it is not continuous at (0, 0).
> plot3d( fxy, 3..3, 3..3 ); Maple can help you with many problems from introductory calculus. For more information, refer to the ?Student[Calculus1] help page. 7.2 Ordinary Diﬀerential Equations Maple provides you with tools for solving, manipulating, and plotting ordinary diﬀerential equations and systems of diﬀerential equations. 7.2 Ordinary Diﬀerential Equations • 241 The dsolve Command
The most commonly used command for investigating the behavior of ordinary diﬀerential equations (ODEs) in Maple is dsolve. You can use this generalpurpose command to obtain both closed form and numerical solutions to a wide variety of ODEs. This is the basic syntax of dsolve. dsolve(eqns, vars ) • eqns is a set of diﬀerential equations and initial values • vars is a set of variables with respect to which dsolve solves The following example is a diﬀerential equation and an initial condition.
> eq := diff(v(t),t)+2*t = 0; eq := (
> ini := v(1) = 5; d v(t)) + 2 t = 0 dt ini := v(1) = 5 Use dsolve to obtain the solution.
> dsolve( {eq, ini}, {v(t)} ); v(t) = −t2 + 6 If you omit some or all of the initial conditions, then dsolve returns a solution containing arbitrary constants of the form _Cnumber .
> eq := diff(y(x),x$2)  y(x) = 1; eq := ( d2 y(x)) − y(x) = 1 dx2 > dsolve( {eq}, {y(x)} ); {y(x) = ex _C2 + e(−x) _C1 − 1} 242 • Chapter 7: Solving Calculus Problems To specify initial conditions for the derivative of a function, use the following notation. D(fcn )(var_value ) = value (D@@n )(fcn )(var_value ) = value • D notation represents the derivative • D@@n notation represents the nth derivative Here is a diﬀerential equation and some initial conditions involving the derivative.
> de1 := diff(y(t),t$2) + 5*diff(y(t),t) + 6*y(t) = 0; de1 := ( d2 d y(t)) + 5 ( y(t)) + 6 y(t) = 0 2 dt dt > ini := y(0)=0, D(y)(0)=1; ini := y(0) = 0, D(y )(0) = 1 Again, use dsolve to ﬁnd the solution.
> dsolve( {de1, ini}, {y(t)} ); y(t) = −e(−3 t) + e(−2 t) Additionally, dsolve may return a solution in parametric form, [x=f(_T), y(x)=g(_T)], where _T is the parameter. The explicit Option Maple may return the solution to a diﬀerential equation in implicit form.
> de2 := diff(y(x),x$2) = (ln(y(x))+1)*diff(y(x),x); de2 := d d2 y(x) = (ln(y(x)) + 1) ( y(x)) dx2 dx > dsolve( {de2}, {y(x)} ); 7.2 Ordinary Diﬀerential Equations
y(x) • 243 {y(x) = _C1 }, 1 d_a − x − _C2 = 0 _a...
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This note was uploaded on 08/27/2012 for the course MATH 1100 taught by Professor Nil during the Spring '12 term at National University of Singapore.
 Spring '12
 NIL
 Math, Division

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