eval alpha110 m1 k1 72 ordinary dierential

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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 general-purpose 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.

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