HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 483

Function LDEC The calculator provides function LDEC (Linear Differential Equation Command) to find the general solution to a linear ODE of any order with constant coefficients, whether it is homogeneous or not. This function requires you to provide two pieces of input: • the right-hand side of the ODE • the characteristic equation of the ODE Both of these inputs must be given in terms of the default independent variable for the calculator's CAS (typically 'X'). The output from the function is the general solution of the ODE. The function LDEC is available through in the CALC/DIFF menu. The examples are shown in the RPN mode, however, translating them to the ALG mode is straightforward. Example 1 - To solve the homogeneous ODE: d3y/dx3-4⋅(d2y/dx2)11⋅(dy/dx)+30⋅y = 0, enter: 0 ` 'X^3-4*X^2-11*X+30' ` LDEC. The solution is: where cC0, cC1, and cC2 are constants of integration. While this result seems very complicated, it can be simplified if we take K1 = (10*cC0-(7+cC1-cC2))/40, K2 = -(6*cC0-(cC1+cC2))/24, and K3 = (15*cC0+(2*cC1-cC2))/15. Then, the solution is y = K1⋅e-3x + K2⋅e5x + K3⋅e2x. The reason why the result provided by LDEC shows such complicated combination of constants is because, internally, to produce the solution, LDEC utilizes Laplace transforms (to be presented later in this chapter), which transform the solution of an ODE into an algebraic solution. The combination Page 16-5