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Math Advanced Math ADVANCED ENGINEERING MATHEMATICS

The corresponding particular solution of the equation 3 ( y + 1 ) d x = 2 x d y , when I . F = ( y + 1 ) x − 4 . The particular solution of differential equation is ( y + 1 ) 2 ( x − 3 ) = c _ . The given differential equation, can be written as: 3 ( y + 1 ) d x = 2 x d y 3 ( y + 1 ) ( y + 1 ) x − 4 d x = 2 x ( y + 1 ) x − 4 d y 3 ( y + 1 ) ( y + 1 ) x 4 d x − 2 ( y + 1 ) x 3 d y = 0 From the given equation the value of M = 3 ( y + 1 ) ( y + 1 ) x 4 , and N = − 2 ( y + 1 ) x 3 . Differentiate the function M , with respect to y . d M d y = d d y [ 3 ( y + 1 ) ( y + 1 ) x 4 ] = 3 × 2 ( y + 1 ) x 4 = 6 ( y + 1 ) x 4 Now differentiate the function N , with respect to x . d N d x = d d x [ − 2 ( y + 1 ) x 3 ] = 2 ( y + 1 ) ( 3 x − 4 ) = 6 ( y + 1 ) x 4 From the above calculation, it can be observed that the value of d M d y = d N d x . Substitute, the value of M = 3 ( y + 1 ) ( y + 1 ) x 4 in the equation u = ∫ M d x + k ( y ) . u = ∫ ( 3 ( y + 1 ) 2 x 4 ) d x + k ( y ) = 3 ( y + 1 ) 2 ∫ 1 x 4 d x + k ( y ) = 3 ( y + 1 ) 2 ( x − 3 3 ) + k ( y ) = − ( y + 1 ) 2 ( x − 3 ) + k ( y ) Substitute the value u = − ( y + 1 ) 2 ( x − 3 ) + k ( y ) in the equation d u d y = N . d d y [ − ( y + 1 ) 2 ( x − 3 ) + k ( y ) ] = − 2 ( y + 1 ) x 3 [ − 2 ( y + 1 ) x 3 + d k d y ] = − 2 ( y + 1 ) x 3 d k d y = 0 k = c Hence the implicit solution of the equation is calculated as: u = 0 − ( y + 1 ) 2 ( x − 3 ) + c = 0 − ( y + 1 ) 2 ( x − 3 ) = − c [ ( y + 1 ) 2 ( x − 3 ) ] = c On further simplification, the following is obtained. − y 2 x = C − x y 2 = x 2 − C x y = x 2 − C x Now check the implicit solution by the differentiation d u = ∂ u ∂ x d x + ∂ u ∂ y d y . d u = ∂ [ ( y + 1 ) 2 ( x − 3 ) ] ∂ x d x + ∂ [ ( y + 1 ) 2 ( x − 3 ) ] ∂ y d y = [ 3 ( y + 1 ) 2 ( x − 4 ) ] d x − 2 ( y + 1 ) x 3 d y = 0 Therefore, the general solution of differential equation is ( y + 1 ) 2 ( x − 3 ) = c _ .

The corresponding particular solution of the equation 3 ( y + 1 ) d x = 2 x d y , when I . F = ( y + 1 ) x − 4 . The particular solution of differential equation is ( y + 1 ) 2 ( x − 3 ) = c _ . The given differential equation, can be written as: 3 ( y + 1 ) d x = 2 x d y 3 ( y + 1 ) ( y + 1 ) x − 4 d x = 2 x ( y + 1 ) x − 4 d y 3 ( y + 1 ) ( y + 1 ) x 4 d x − 2 ( y + 1 ) x 3 d y = 0 From the given equation the value of M = 3 ( y + 1 ) ( y + 1 ) x 4 , and N = − 2 ( y + 1 ) x 3 . Differentiate the function M , with respect to y . d M d y = d d y [ 3 ( y + 1 ) ( y + 1 ) x 4 ] = 3 × 2 ( y + 1 ) x 4 = 6 ( y + 1 ) x 4 Now differentiate the function N , with respect to x . d N d x = d d x [ − 2 ( y + 1 ) x 3 ] = 2 ( y + 1 ) ( 3 x − 4 ) = 6 ( y + 1 ) x 4 From the above calculation, it can be observed that the value of d M d y = d N d x . Substitute, the value of M = 3 ( y + 1 ) ( y + 1 ) x 4 in the equation u = ∫ M d x + k ( y ) . u = ∫ ( 3 ( y + 1 ) 2 x 4 ) d x + k ( y ) = 3 ( y + 1 ) 2 ∫ 1 x 4 d x + k ( y ) = 3 ( y + 1 ) 2 ( x − 3 3 ) + k ( y ) = − ( y + 1 ) 2 ( x − 3 ) + k ( y ) Substitute the value u = − ( y + 1 ) 2 ( x − 3 ) + k ( y ) in the equation d u d y = N . d d y [ − ( y + 1 ) 2 ( x − 3 ) + k ( y ) ] = − 2 ( y + 1 ) x 3 [ − 2 ( y + 1 ) x 3 + d k d y ] = − 2 ( y + 1 ) x 3 d k d y = 0 k = c Hence the implicit solution of the equation is calculated as: u = 0 − ( y + 1 ) 2 ( x − 3 ) + c = 0 − ( y + 1 ) 2 ( x − 3 ) = − c [ ( y + 1 ) 2 ( x − 3 ) ] = c On further simplification, the following is obtained. − y 2 x = C − x y 2 = x 2 − C x y = x 2 − C x Now check the implicit solution by the differentiation d u = ∂ u ∂ x d x + ∂ u ∂ y d y . d u = ∂ [ ( y + 1 ) 2 ( x − 3 ) ] ∂ x d x + ∂ [ ( y + 1 ) 2 ( x − 3 ) ] ∂ y d y = [ 3 ( y + 1 ) 2 ( x − 4 ) ] d x − 2 ( y + 1 ) x 3 d y = 0 Therefore, the general solution of differential equation is ( y + 1 ) 2 ( x − 3 ) = c _ .

ADVANCED ENGINEERING MATHEMATICS

ADVANCED ENGINEERING MATHEMATICS

10th Edition

ISBN: 9781119664697

Author: Kreyszig

Publisher: WILEY

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ADVANCED ENGINEERING MATHEMATICS

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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