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), (38) provided one solution y₁ of the corresponding homogeneous equation is known. Let y = v(t)yi(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 equation for v'. Solving this equation, integrating the result, and then multiplying by y₁(t) leads to the general solution of the first equation. Use the method above to solve the differential equation t²y" - 2ty' + 2y = 9t², t > 0, y₁(t) = t. NOTE: Use c₁ and ca 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), (38) provided one solution y₁ of the corresponding homogeneous equation is known. Let y = v(t)yi(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 equation for v'. Solving this equation, integrating the result, and then multiplying by y₁(t) leads to the general solution of the first equation. Use the method above to solve the differential equation t²y" - 2ty' + 2y = 9t², t > 0, y₁(t) = t. NOTE: Use c₁ and ca 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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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),
(38)
provided one solution y₁ of the corresponding homogeneous equation
is known. Let y = v(t)yi(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 equation for v'. Solving this
equation, integrating the result, and then multiplying by y₁(t) leads
to the general solution of the first equation.
Use the method above to solve the differential equation
t²y" - 2ty' + 2y = 9t², t > 0, y₁(t) = t.
NOTE: Use c₁ and ca as arbitrary constants.
y(t)
=
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