The method of reduction of order (Section 3.4) can also be used for the nonhomogeneous equation y/" + p(t)y' + q(t)y = g(t), provided one solution y1 of the corresponding homogeneous equation is known. Let y = v(t)y1(t). It can be shown that y satisfies (38) equation (38) if v is a solution of y1(t)u" + (2y (t) + p(t)y1(t))v' = g(t). Equation (39) is a first order linear equation for v'. Solving this equation, integrating the result, and then multiplying by y1(t) leads (39) to the general solution of the first equation. Use the method above to solve the differential equation ty" – 2ty + 2y = 14t², t > 0, y1(t) = t. - NOTE: Use c; and cz as arbitrary constants. y(t) :
The method of reduction of order (Section 3.4) can also be used for the nonhomogeneous equation y/" + p(t)y' + q(t)y = g(t), provided one solution y1 of the corresponding homogeneous equation is known. Let y = v(t)y1(t). It can be shown that y satisfies (38) equation (38) if v is a solution of y1(t)u" + (2y (t) + p(t)y1(t))v' = g(t). Equation (39) is a first order linear equation for v'. Solving this equation, integrating the result, and then multiplying by y1(t) leads (39) to the general solution of the first equation. Use the method above to solve the differential equation ty" – 2ty + 2y = 14t², t > 0, y1(t) = t. - NOTE: Use c; and cz as arbitrary constants. y(t) :
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
\[ y'' + p(t)y' + q(t)y = g(t), \tag{38} \]
provided one solution \( y_1 \) of the corresponding homogeneous equation is known. Let \( y = v(t)y_1(t) \). It can be shown that \( y \) satisfies equation (38) if \( v \) is a solution of
\[ y_1(t)v'' + \left(2y_1'(t) + p(t)y_1(t)\right)v' = g(t). \tag{39} \]
Equation (39) is a first order linear equation for \( v' \). Solving this equation, integrating the result, and then multiplying by \( y_1(t) \) leads to the general solution of the first equation.
Use the method above to solve the differential equation
\[ t^2y'' - 2ty' + 2y = 14t^2, \quad t > 0, \quad y_1(t) = t. \]
**NOTE:** Use \( c_1 \) and \( c_2 \) as arbitrary constants.
\[ y(t) = \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0bff7fb0-7afb-4d12-af99-7cba91505041%2F803ebab7-81a6-4902-b3de-cadb442ad4b6%2F9mwhvmo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The method of reduction of order (Section 3.4) can also be used for the nonhomogeneous equation
\[ y'' + p(t)y' + q(t)y = g(t), \tag{38} \]
provided one solution \( y_1 \) of the corresponding homogeneous equation is known. Let \( y = v(t)y_1(t) \). It can be shown that \( y \) satisfies equation (38) if \( v \) is a solution of
\[ y_1(t)v'' + \left(2y_1'(t) + p(t)y_1(t)\right)v' = g(t). \tag{39} \]
Equation (39) is a first order linear equation for \( v' \). Solving this equation, integrating the result, and then multiplying by \( y_1(t) \) leads to the general solution of the first equation.
Use the method above to solve the differential equation
\[ t^2y'' - 2ty' + 2y = 14t^2, \quad t > 0, \quad y_1(t) = t. \]
**NOTE:** Use \( c_1 \) and \( c_2 \) as arbitrary constants.
\[ y(t) = \]
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