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HP 48gII Differential Equations, Chapter, The CALC/DIFF menu, Solution to linear and non-linearequations

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Chapter 14

Differential Equations

In this Chapter we present examples of solving ordinary differential equations (ODE) using calculator functions. A differential equation is an equation involving derivatives of the independent variable. In most cases, we seek the dependent function that satisfies the differential equation.

The CALC/DIFF menu

The DIFFERENTIAL EQNS.. sub-menu within the CALC („Ö) menu provides functions for the solution of differential equations. The menu is listed below with system flag 117 set to CHOOSE boxes:

These functions are briefly described next. They will be described in more detail in later parts of this Chapter.

DESOLVE: Differential Equation SOLVEr, solves differential equations, when possible

ILAP: Inverse LAPlace transform, L-1[F(s)] = f(t)

LAP: LAPlace transform, L[f(t)]=F(s)

LDEC: Linear Differential Equation Command

Solution to linear and non-linear equations

An equation in which the dependent variable and all its pertinent derivatives are of the first degree is referred to as a linear differential equation. Otherwise, the equation is said to be non-linear.

Page 14-1

Contents

graphing calculator Page Preface Table of Contents Chapter 3 – Calculations with real numbers Chapter 4 – Calculations with complex numbers Chapter 5 – Algebraic and arithmetic operations Chapter 6 – Solution to equations Chapter 7 – Operations with lists The SEQ function The MAP function Chapter 8 – Vectors Entering vectors Typing vectors in the stack Chapter 10 – Graphics Chapter 11 – Calculus Applications Chapter 12 – Multi-variateCalculus Applications Chapter 13 – Vector Analysis Applications Chapter 14 – Differential Equations Chapter 15 – Probability Distributions Chapter 16 – Statistical Applications Chapter 17 – Numbers in Different Bases Warranty – W-1 Chapter Getting started Turning the calculator on and off Adjusting the display contrast Contents of the calculator’s display @EDIT @VIEW @@RCL @@ @@STO@ ! PURGE !CLEAR ABCDEF Menus The TOOL menu @EDIT A EDIT the contents of a variable (see Chapter 2 in this Guide and Chapter 2 and Appendix L in the user's guide for more information on editing) B VIEW the contents of a variable Page ~„p ~…p SYMB Selecting calculator modes !!@@OK#@ F Operating Mode @CHOOSE !!@@OK#@ !@.#*+-/R Q¸Ü‚Oš™˜—` 1./3.*3 /23.Q3™™™+!¸2.5` R!Ü3.*!Ü5 1/3.*3.™ /23.Q3+!¸2.5` 3+2` 3`2`+ 123`32 424`2Q 3√2727`R3@» 23` H@)FLAGS —„—„—„—@@CHK@ Number Format and decimal dot or comma •Standard format: •Fixed format with decimals: Fixed Fix Press the !!@@OK#@ soft menu key to complete the selection: Page Page @CHECK Angle Measure Coordinate System Selecting CAS settings •Press the Hbutton to activate the CALCULATOR MODES input form To change CAS settings press the @@ CAS@@ š™˜— _Numeric, _Approx, _Complex, _Verbose, _Step/Step, _Incr Pow Explanation of CAS settings •Indep var: The independent variable for CAS applications. Typically, VX = ‘X’ Modulo Numeric @@CHK@@ Textbook Stack: _Small, _Full page _Indent Ft8_0:system Selecting properties of the line editor Edit _Small Changes font size to small Selecting properties of the Stack ‚O…Á0™„è™„¸\x™x` Selecting properties of the equation writer (EQW) References Page Introducing the calculator 5*„Ü1+1/7.5™ „ÜR3-2Q3 ³5*„Ü1+1/7.5™ „ÜR3-2Q3` µ„î …³5*„Ü1+1/7.5™ ™…ï Creating algebraic expressions 2L 1 + 2*~l*R„Ü1+~„x/~r™/„ Ü~r+~„y™+2*~l/~„b Using the Equation Writer (EQW) to create expressions 5/5+2 The result is the expression *„Ü5+1/3 ƒƒƒ„ìQ2 NOTE +1/3  x + 2∝ ⋅ ∆y  λ + e Use the following keystrokes: 2/R3™™*~‚n+„¸\~‚m ™™*‚¹~„x+2*~‚m*~‚c ~„y———/~‚tQ1/3 Organizing data in the calculator The HOME directory Subdirectories Variables Typing variable names ³~~math1~` ³~~m„a„t„h~` ³~~m„~at„h~` Creating variables Name A12: 3V5K~a12` Q:³~„r/„Ü ~„m+~„r™™K~q` R:„Ô3‚í2‚í1™K~r` 3+5*„¥K~„z1` p1: ‚å‚é~„r³„ì* ~„rQ2™™™K~„p1` I\@@OK@@ 0.25\`~‚a` 3V5³~a12`K Q:³~„r/„Ü ~„m+~„r™™³~q`K R:„Ô3#2#1™³K p1: ‚å‚é~„r³„ì* ~„rQ2™™™³~„p1™`K Checking variables contents J@@ @@ `@@@R@@ `@@@ @@@ ` At this point, the screen looks like this: J‚@@ @@‚@@ @@ ‚@@@R@@ ‚@@@ @@ ‚@@A @@ Deleting variables „¡@@OK@@ Using function PURGE in the stack in Algebraic mode I@PURGE@ J@@ @@ ` I@PURGE@ „ä³J@@@R!@@ ™‚í³J@@@ !@@ J„ä³@@@R!@@ ™³@@@ !@@ ` I@PURGE@ UNDO and CMD functions CHOOSE boxes vs. Soft MENU @@OK@@ ˜˜˜˜ @@OK@@ H@FLAGS! ——————— @CHECK @@MEM@@) @@DIR@@) @ORDER Calculations with real numbers 6.3` 8.5- 4.2` 2.5* 2.3` 4.5 3.7#5.2+ 6.3#8.5- 4.2#2.5* 2.3#4.5 „Ü5+3.2™/„Ü7- 2.2` 5`3.2`+7`2.2` ³„Ü5+3.2 „Ê\2.32` 2.32\„Ê „º\2.3` 2.3\„º R123.4` ‚»3‚í27` 27`3‚» ‚Ã2.45` „Â\2.3` 2.45‚Ã 2.3\„Â 2.45`‚¹ 2.3\`„¸ S30` T45` U135` 30S Real number functions in the MTH menu Using calculator menus: We will describe in detail the use of the 4. HYPERBOLIC Hyperbolic functions and their inverses „´4@@OK@@ 5@@OK@@ 2.5` 2.5`„´4@@OK@@ 5@@OK@@ @@H) P@ @@TA H@ Note: „´@@H P@ @@TA H@ 2.5` 2.5`„´@@H) P@ @@TA H@ Operations with units Page @)SPEED Pressing the soft menu key @)U ITS will take you back to the UNITS menu Note: Use the Lkey or the „«keystroke sequence to navigate through the menus Available units Attaching units to numbers 5‚Ý‚Û8@@OK@@ @@OK@@ ` Note 5‚Û8@@OK@@ @@OK@@ 5‚Ý‚ÛL@)@FORCE @@@ @@ ` 123‚Ý~„p~„m Operations with units Page Unit conversions Physical constants in the calculator ~~conlib~` ‚N~c @E GL @U ITS To copy the value of Vm to the stack, select the variable name, and STK @ UIT@ tagged value tag „à³~h„Ü~„x™‚Å ‚¹~„x+1™+„¸~„x` The screen will look like this: @@@H@@ ‚@@@H@@ @@@H@@@ „Ü2` Reference Calculations with complex numbers Entering complex numbers „Ü3.5‚í\1.2` x+iy 3.5-1.2*‚¥` In RPN mode, these numbers will be entered using the following keystrokes: ~‚6 Simple operations with complex numbers Complex numbers can be combined using the four fundamental operations The CMPLX menus CMPLX menu through the MTH menu CHOOSE boxes „´9@@OK@@ CMPLX menu in the keyboard Functions applied to complex numbers Function DROITE: equation of a straight line Page Algebraic and arithmetic operations Simple operations with algebraic objects ³„ì*~rQ™K~a1` ³„ì*~rQ`~a1K @@A @@ @@A @@ +@@A @@ ` @@A @@ +@@A @@ ` @@A @@ 8@@A @@ ` @@A @@ /@@A @@ ` ‚¹@@A @@ „¸v Functions in the ALG menu IL @)HELP@ ` To complete the operation press @@OK@@. Here is the help screen for function COLLECT: For example, for function SUBST, we find the following CAS help facility entry: ³„¸+~x+~y` Operations with transcendental functions Expansion and factoring using log-expfunctions The „Ðproduces the following menu: Expansion and factoring using trigonometric functions The TRIG menu, triggered by using ‚Ñ, shows the following functions: IL@HELP Functions in the ARITHMETIC menu FACTORS:SIMP2: Polynomials The HORNER function HORNER(‘X^3+2*X^2-3*X+1’,2)= {‘X^2+4*X+5, 2, 11} i.e., X3+2X2-3X+1= (X2+4X+5)(X-2)+11.Also The variable The PCOEF function The PROOT function The QUOTIENT and REMAINDER functions The PEVAL function Fractions The SIMP2 function The PROPFRAC function The PARTFRAC function The FCOEF function The FROOTS function Step-by-stepoperations with polynomials and fractions X 3 − 5X 2 + 3X − Page Solution to equations Function SOLVE Page Function SOLVEVX Function ZEROS Numerical solver menu Notes: Polynomial Equations ‚Ï˜˜@@OK@@ „Ô3‚í2‚í0 ‚í1\‚í1@@OK@@ @SOLVE@ ˜„Ô1‚í5 ‚í2\‚í4@@OK@@ „Ô1‚í5 —@SM @ ˜„Ô1‚í3 ‚í2\‚í1@@OK@@ ˜@S M @ Financial calculations Solving equations with one unknown through NUM.SLV ³„¸~„x™-S„ì*~„x/3™‚Å0™ K~e~q` ‚Ï@@OK@@ 3\@@@OK@@˜@SOLVE@ Solution to simultaneous equations with MSLV @ECHO Page Operations with lists Addition, subtraction, multiplication, division Functions applied to lists ABS INVERSE (1/x) Lists of complex numbers Lists of algebraic objects The MTH/LIST menu Page The SEQ function The MAP function Vectors Storing vectors into variables in the stack Using the matrix writer (MTRW) to enter vectors @EDIT! @VEC„ ←WID @WID→ @GO→„ @GO↓ @VEC@@ WID @WID @GO 3`5`2`` @ ROW@ @ ROW @ COL@ STK@@ @GOTO@ @GOTO@ 3@@OK@@ 3@@OK@@ @@OK@@ Simple operations with vectors Changing sign Addition, subtraction Multiplication by a scalar, and division by a scalar Absolute value function The MTH/VECTOR menu Magnitude Dot product Cross product Matrices and linear algebra H@)DISP Typing in the matrix directly into the stack „Ô 2.5\‚í4.2‚í2™ ‚í „Ô .3‚í1.9‚í2.8™ ‚í „Ô 2‚í.1\‚í.5 ‚í Operations with matrices '`K Addition and subtraction `+ A A` `+ A ` ` Multiplication for Page Characterizing a matrix (The matrix NORM menu) Function DET Function DET calculates the determinant of a square matrix. For example Function TRACE tr(A) = ∑aii an2 , and Solution with the inverse matrix Solution by “division” of matrices Graphics Plotting an expression of the form y = f(x) f (x) = 1 exp(− x 2 ) L@@@OK@@@ @ADD @AUTO @ERASE @DRAW •To see labels: @EDIT L@LA EL @ME U LL@)PICT @TRACE @@ @@ Next, press To accept the changes made to the PLOT SETUP screen press and the increment (Step). Enter the following: 5\@@@OK@@@ 0.5@@@OK@@@ 0.5@@@OK@@@ @CHK Out Decimal, Integer Trig With the option zoom in @CHOOS š™—˜ •When done, press @E IT •Press @CA CL to return to the PLOT WINDOW environment • Change the Step data to read: Step Indep: 20 Depnd: •Press @ERASE @DRAW to see the surface plot. Sample views: •Press @CA CL to return to PLOT WINDOW Page Calculus Applications ‚N~„l Functions DERIV and DERVX ∫f (x)dx = F(x) + C Definite integrals ∫ab f (x)dx = F(b) − F(a) Infinite series (xo ) f (x) = ∑ ⋅ (x − xo ) n If the value x0 = 0, the series is referred to as a Maclaurin’s series Page Page Multi-variateCalculus Applications dydx Vector Analysis Applications The del operator Divergence divF Curl curl Differential Equations Function LDEC Function DESOLVE y(x) = 5 ⋅ exp(−x3 / 3) ⋅ (∫ exp(x3 / 3) ⋅ dx + C0 ) The variable ODETYPE @ODET J @ODET Laplace Transforms Laplace transform and inverses in the calculator L{ f (t)} = F (s) = ∫0∞ f (t) ⋅ e−st dt 2inπt „(hold) §`J@)CASDI `2K@PERIOD ` Page Probability Distributions ~‚2 Random numbers The MTH/PROB menu - part The Normal distribution The Student-tdistribution The Chi-squaredistribution The F distribution Statistical Applications CHK@ @@@OK@@ Obtaining frequency distributions ‚Ù˜@@@OK@@@ ‚Ù@@@OK@@@ Select the program •Change X-Minto -8,Bin Count to 8, and Bin Width to 2, then press @@@OK@@@ The bins for this frequency distribution will be: -8to -6, -6to -4,… 23, 22, 22, 17, 26, 15, 20 Fitting data to a function y = f(x) ‚Ù˜˜@@@OK@@@ Obtaining additional summary statistics •To access the summary stats… option, use: ‚Ù˜˜˜@@@OK@@@ Confidence intervals 6. Conf Interval ‚Ù— T-INT: Press @GRAPH to see a graphical display of the confidence interval information: Hypothesis testing 6. Conf Interval ‚Ù——@@@OK@@@ These options are interpreted as in the confidence interval applications: ‚Ù—— Page Numbers in Different Bases @HE Warranty Service Europe Country : Telephone numbers Austria Page Regulatory information