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The ODE corresponds to dropping a stone is the familiar law of free fall y = 1 2 g t 2 . When dropping a stone of iron ball, the air resistance is negligible and the acceleration of the motion is constant where g = 9.8 m/s 2 = 32 ft/s 2 , is known as the acceleration of gravity. Let the distance fallen at time t be y t . Then the velocity is given by y ′ t and the acceleration is given by g = y ″ t . Integrate g = y ″ t and find the expression for velocity. ∫ d 2 y d t 2 = ∫ g d t ∫ d d y d t = ∫ g d t d y d t = g t + c y ′ = g t + c The motion starts from the rest and hence v = 0 when t = 0 . This implies the initial condition, y ′ 0 = 0 . Substitute t = 0 and y ′ t = 0 in y ′ = g t + c . y ′ = g t + c 0 = g 0 + c c = 0 Hence, y ′ t = g t . Integrate y ′ t = g t and find the expression for y . ∫ d y d t = ∫ g t d t y = g t 2 2 + C y = g t 2 2 + C The motion starts from the rest and y = 0 when t = 0 . Substitute t = 0 and y t = 0 in y = g t 2 2 + C . y = g t 2 2 + C 0 = g 0 2 + C C = 0 Therefore, the law of free fall y t = 1 2 g t 2 is obtained.

The ODE corresponds to dropping a stone is the familiar law of free fall y = 1 2 g t 2 . When dropping a stone of iron ball, the air resistance is negligible and the acceleration of the motion is constant where g = 9.8 m/s 2 = 32 ft/s 2 , is known as the acceleration of gravity. Let the distance fallen at time t be y t . Then the velocity is given by y ′ t and the acceleration is given by g = y ″ t . Integrate g = y ″ t and find the expression for velocity. ∫ d 2 y d t 2 = ∫ g d t ∫ d d y d t = ∫ g d t d y d t = g t + c y ′ = g t + c The motion starts from the rest and hence v = 0 when t = 0 . This implies the initial condition, y ′ 0 = 0 . Substitute t = 0 and y ′ t = 0 in y ′ = g t + c . y ′ = g t + c 0 = g 0 + c c = 0 Hence, y ′ t = g t . Integrate y ′ t = g t and find the expression for y . ∫ d y d t = ∫ g t d t y = g t 2 2 + C y = g t 2 2 + C The motion starts from the rest and y = 0 when t = 0 . Substitute t = 0 and y t = 0 in y = g t 2 2 + C . y = g t 2 2 + C 0 = g 0 2 + C C = 0 Therefore, the law of free fall y t = 1 2 g t 2 is obtained.

Advanced Engineering Mathematics

Advanced Engineering Mathematics

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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Ch. 1.1 - Prob. 1P Ch. 1.1 - Prob. 2P Ch. 1.1 - Prob. 3P Ch. 1.1 - Prob. 4P Ch. 1.1 - Prob. 5P Ch. 1.1 - Prob. 6P Ch. 1.1 - Prob. 7P Ch. 1.1 - Prob. 8P Ch. 1.1 - Prob. 9P Ch. 1.1 - Prob. 10P

Ch. 1.1 - Prob. 11P Ch. 1.1 - Prob. 12P Ch. 1.1 - Prob. 13P Ch. 1.1 - Prob. 14P Ch. 1.1 - 9–15 VERIFICATION. INITIAL VALUE PROBLEM...Ch. 1.1 - Prob. 16P Ch. 1.1 - Half-life. The half-life measures exponential...Ch. 1.1 - Half-life. Radium has a half-life of about 3.6...Ch. 1.1 - Prob. 19P Ch. 1.1 - Exponential decay. Subsonic flight. The efficiency...Ch. 1.2 - DIRECTION FIELDS, SOLUTION CURVES Graph a...Ch. 1.2 - 1–8 DIRECTION FIELDS, SOLUTION CURVES Graph a...Ch. 1.2 - DIRECTION FIELDS, SOLUTION CURVES Graph a...Ch. 1.2 - Prob. 4P Ch. 1.2 - DIRECTION FIELDS, SOLUTION CURVES Graph a...Ch. 1.2 - Prob. 6P Ch. 1.2 - DIRECTION FIELDS, SOLUTION CURVES Graph a...Ch. 1.2 - Prob. 8P Ch. 1.2 - Prob. 9P Ch. 1.2 - Prob. 10P Ch. 1.2 - Autonomous ODE. This means an ODE not showing x...Ch. 1.2 - Model the motion of a body B on a straight line...Ch. 1.2 - Prob. 13P Ch. 1.2 - Prob. 14P Ch. 1.2 - Prob. 15P Ch. 1.2 - Prob. 16P Ch. 1.2 - EULER’S METHOD This is the simplest method to...Ch. 1.2 - EULER’S METHOD This is the simplest method to...Ch. 1.2 - EULER’S METHOD This is the simplest method to...Ch. 1.2 - EULER’S METHOD This is the simplest method to...Ch. 1.3 - Prob. 1P Ch. 1.3 - Prob. 2P Ch. 1.3 - GENERAL SOLUTION Find a general solution. Show the...Ch. 1.3 - GENERAL SOLUTION Find a general solution. Show the...Ch. 1.3 - GENERAL SOLUTION Find a general solution. Show the...Ch. 1.3 - GENERAL SOLUTION Find a general solution. Show the...Ch. 1.3 - GENERAL SOLUTION Find a general solution. Show the...Ch. 1.3 - GENERAL SOLUTION Find a general solution. Show the...Ch. 1.3 - GENERAL SOLUTION Find a general solution. Show the...Ch. 1.3 - GENERAL SOLUTION Find a general solution. Show the...Ch. 1.3 - INITIAL VALUE PROBLEMS (IVPs) Solve the IVP. Show...Ch. 1.3 - INITIAL VALUE PROBLEMS (IVPs) Solve the IVP. Show...Ch. 1.3 - INITIAL VALUE PROBLEMS (IVPs) Solve the IVP. Show...Ch. 1.3 - INITIAL VALUE PROBLEMS (IVPs) Solve the IVP. Show...Ch. 1.3 - INITIAL VALUE PROBLEMS (IVPs) Solve the IVP. Show...Ch. 1.3 - INITIAL VALUE PROBLEMS (IVPs) Solve the IVP. Show...Ch. 1.3 - Prob. 17P Ch. 1.3 - Prob. 18P Ch. 1.3 - INITIAL VALUE PROBLEMS (IVPs) Solve the IVP. Show...Ch. 1.3 - Prob. 20P Ch. 1.3 - Radiocarbon dating. What should be the content...Ch. 1.3 - Prob. 22P Ch. 1.3 - Prob. 23P Ch. 1.3 - Prob. 24P Ch. 1.3 - Prob. 25P Ch. 1.3 - Prob. 26P Ch. 1.3 - Prob. 27P Ch. 1.3 - Prob. 28P Ch. 1.3 - Prob. 29P Ch. 1.3 - Prob. 30P Ch. 1.3 - Prob. 31P Ch. 1.3 - Prob. 32P Ch. 1.3 - Prob. 33P Ch. 1.3 - Prob. 36P Ch. 1.4 - Prob. 1P Ch. 1.4 - Prob. 2P Ch. 1.4 - Prob. 3P Ch. 1.4 - Prob. 4P Ch. 1.4 - Prob. 5P Ch. 1.4 - Prob. 6P Ch. 1.4 - Prob. 7P Ch. 1.4 - Prob. 8P Ch. 1.4 - Prob. 9P Ch. 1.4 - ODEs. INTEGRATING FACTORS Test for exactness. If...Ch. 1.4 - ODEs. INTEGRATING FACTORS Test for exactness. If...Ch. 1.4 - ODEs. INTEGRATING FACTORS Test for exactness. If...Ch. 1.4 - ODEs. INTEGRATING FACTORS Test for exactness. If...Ch. 1.4 - ODEs. INTEGRATING FACTORS Test for exactness. If...Ch. 1.4 - Exactness. Under what conditions for the constants...Ch. 1.4 - Prob. 17P Ch. 1.4 - Prob. 18P Ch. 1.5 - CAUTION! Show that e−ln x = 1/x (not −x) and...Ch. 1.5 - Prob. 2P Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS Find the...Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS Find the...Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS Find the...Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS Find the...Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS 7. xy′ =...Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS Find the...Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS 9. Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS Find the...Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS Find the...Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS Find the...Ch. 1.5 - GENERAL SOLUTION. INITIAL VALUE PROBLEMS Find the...Ch. 1.5 - Prob. 14P Ch. 1.5 - Prob. 15P Ch. 1.5 - Prob. 16P Ch. 1.5 - Prob. 17P Ch. 1.5 - Prob. 18P Ch. 1.5 - Prob. 19P Ch. 1.5 - GENERAL PROPERTIES OF LINEAR ODEs These properties...Ch. 1.5 - Prob. 21P Ch. 1.5 - NONLINEAR ODEs Using a method of this section or...Ch. 1.5 - NONLINEAR ODEs Using a method of this section or...Ch. 1.5 - NONLINEAR ODEs Using a method of this section or...Ch. 1.5 - NONLINEAR ODEs Using a method of this section or...Ch. 1.5 - NONLINEAR ODEs Using a method of this section or...Ch. 1.5 - NONLINEAR ODEs Using a method of this section or...Ch. 1.5 - NONLINEAR ODEs Using a method of this section or...Ch. 1.5 - Prob. 29P Ch. 1.5 - MODELING. FURTHER APPLICATIONS 31. Newton’s law of...Ch. 1.5 - Prob. 32P Ch. 1.5 - MODELING. FURTHER APPLICATIONS 33. Drug injection....Ch. 1.5 - MODELING. FURTHER APPLICATIONS 34. Epidemics. A...Ch. 1.5 - MODELING. FURTHER APPLICATIONS 35. Lake Erie. Lake...Ch. 1.5 - MODELING. FURTHER APPLICATIONS 36. Harvesting...Ch. 1.5 - Prob. 37P Ch. 1.5 - Prob. 38P Ch. 1.5 - Prob. 39P Ch. 1.5 - Prob. 40P Ch. 1.6 - Represent the given family of curves in the form...Ch. 1.6 - Prob. 2P Ch. 1.6 - Represent the given family of curves in the form...Ch. 1.6 - ORTHOGONAL TRAJECTORIES (OTs) Sketch or graph some...Ch. 1.6 - ORTHOGONAL TRAJECTORIES (OTs) Sketch or graph some...Ch. 1.6 - ORTHOGONAL TRAJECTORIES (OTs) Sketch or graph some...Ch. 1.6 - ORTHOGONAL TRAJECTORIES (OTs) Sketch or graph some...Ch. 1.6 - ORTHOGONAL TRAJECTORIES (OTs) Sketch or graph some...Ch. 1.6 - ORTHOGONAL TRAJECTORIES (OTs) Sketch or graph some...Ch. 1.6 - ORTHOGONAL TRAJECTORIES (OTs) Sketch or graph some...Ch. 1.6 - APPLICATIONS, EXTENSIONS 11. Electric field. Let...Ch. 1.6 - Electric field. The lines of electric force of two...Ch. 1.6 - Prob. 13P Ch. 1.6 - Conic sections. Find the conditions under which...Ch. 1.6 - Prob. 15P Ch. 1.6 - Prob. 16P Ch. 1.7 - Prob. 1P Ch. 1.7 - Existence? Does the initial value problem (x −...Ch. 1.7 - Vertical strip. If the assumptions of Theorems 1...Ch. 1.7 - Change of initial condition. What happens in Prob....Ch. 1.7 - Prob. 5P Ch. 1.7 - Maximum α. What is the largest possible α in...Ch. 1.7 - Prob. 8P Ch. 1.7 - Common points. Can two solution curves of the same...Ch. 1.7 - Three possible cases. Find all initial conditions...Ch. 1 - Prob. 1RQ Ch. 1 - Prob. 2RQ Ch. 1 - Does every first-order ODE have a solution? A...Ch. 1 - What is a direction field? A numeric method for...Ch. 1 - What is an exact ODE? Is f(x) dx + g(y) dy = 0...Ch. 1 - Prob. 6RQ Ch. 1 - What other solution methods did we consider in...Ch. 1 - Can an ODE sometimes be solved by several methods?...Ch. 1 - Prob. 9RQ Ch. 1 - Prob. 10RQ Ch. 1 - Prob. 11RQ Ch. 1 - Prob. 12RQ Ch. 1 - Prob. 13RQ Ch. 1 - Prob. 14RQ Ch. 1 - Prob. 15RQ Ch. 1 - DIRECTION FIELD: NUMERIC SOLUTION Graph a...Ch. 1 - Prob. 17RQ Ch. 1 - Prob. 18RQ Ch. 1 - Prob. 19RQ Ch. 1 - Prob. 20RQ Ch. 1 - Prob. 21RQ Ch. 1 - Prob. 22RQ Ch. 1 - Prob. 23RQ Ch. 1 - Prob. 24RQ Ch. 1 - Prob. 25RQ Ch. 1 - Prob. 26RQ Ch. 1 - Prob. 27RQ Ch. 1 - Prob. 28RQ Ch. 1 - Half-life. If in a reactor, uranium loses 10% of...Ch. 1 - Prob. 30RQ

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