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ition and a particular solution. e the method above to solve the differential equation ty" – (1+t)y' + y = 7t²e", t > 0, y1(t) = 1+t %3D

ition and a particular solution. e the method above to solve the differential equation ty" – (1+t)y' + y = 7t²e", t > 0, y1(t) = 1+t %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The method of reduction of order (Section 3.4) can also be used for the nonhomogeneous equation y" + p(t)y' + q(t)Y = 9(t), (38) provided one solution y1 of the corresponding homogeneous equation is known. Let y = v(t)y1(t). It can be shown that y satisfies equation (38) if v is a solution of Y1(t)v" + (2y (t) + p(t)y1(t))v' = g(t). (39) Equation (39) is a first-order linear differential equation for v'. By solving equation (39) for v', integrating the result to find v, and then multiplying by yı(t), we can find the general solution of equation (38). This method simultaneously finds both the second homogeneous solution and a particular solution.
Transcribed Image Text:The method of reduction of order (Section 3.4) can also be used for the nonhomogeneous equation y" + p(t)y' + q(t)Y = 9(t), (38) provided one solution y1 of the corresponding homogeneous equation is known. Let y = v(t)y1(t). It can be shown that y satisfies equation (38) if v is a solution of Y1(t)v" + (2y (t) + p(t)y1(t))v' = g(t). (39) Equation (39) is a first-order linear differential equation for v'. By solving equation (39) for v', integrating the result to find v, and then multiplying by yı(t), we can find the general solution of equation (38). This method simultaneously finds both the second homogeneous solution and a particular solution.
Equation (39) is a first-order linear differential equation for v'. By solving equation (39) for v', integrating the result to find v, multiplying by Yı(t), we can find the general solution of equation (38). This method simultaneously finds both the second homogeneous solution and a particular solution. and then Use the method above to solve the differential equation ty" – (1+ t)y' + y = 7t°e*, t > 0, Yı(t) = 1+ t. NOTE: Use cı, c2, ... for the constants of integration. y(t)
Transcribed Image Text:Equation (39) is a first-order linear differential equation for v'. By solving equation (39) for v', integrating the result to find v, multiplying by Yı(t), we can find the general solution of equation (38). This method simultaneously finds both the second homogeneous solution and a particular solution. and then Use the method above to solve the differential equation ty" – (1+ t)y' + y = 7t°e*, t > 0, Yı(t) = 1+ t. NOTE: Use cı, c2, ... for the constants of integration. y(t)
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