Verify that the given functions y₁(t) = t and y₂(t) = tet satisfy the corresponding homogeneous equation; then find a particular solution of the nonhomogeneous equation that does not involve any terms from the homogeneous solution. t²y" - t (6t+2)y' + (6t + 2)y = 5t³, t > 0. NOTE: Use t as the independent variable. Y(t) = eTextbook and Media Hint Assistance Used If the functions p, q, and g are continuous on an open interval I and if the functions y, and y2 are a fundamental set of solutions of the homogeneous equation y" +p(t)y + g(t)y = 0 corresponding to the nonhomogeneous equation y" + p(t)y + q(t)y = g(t). then a particular solution of the nonhomogeneous equation is Y(t) = - y₁ (1) 2)(a)_ds + y₂(1)(a) ds, where to is any W(Y1J2) fo W(y1.92)(s) conveniently chosen point in I. The general solution is y = €₁Y₁ (1) + €₂Y₂(1) + Y(t).
Verify that the given functions y₁(t) = t and y₂(t) = tet satisfy the corresponding homogeneous equation; then find a particular solution of the nonhomogeneous equation that does not involve any terms from the homogeneous solution. t²y" - t (6t+2)y' + (6t + 2)y = 5t³, t > 0. NOTE: Use t as the independent variable. Y(t) = eTextbook and Media Hint Assistance Used If the functions p, q, and g are continuous on an open interval I and if the functions y, and y2 are a fundamental set of solutions of the homogeneous equation y" +p(t)y + g(t)y = 0 corresponding to the nonhomogeneous equation y" + p(t)y + q(t)y = g(t). then a particular solution of the nonhomogeneous equation is Y(t) = - y₁ (1) 2)(a)_ds + y₂(1)(a) ds, where to is any W(Y1J2) fo W(y1.92)(s) conveniently chosen point in I. The general solution is y = €₁Y₁ (1) + €₂Y₂(1) + 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:Verify that the given functions y₁(t) = t and y₂(t) = tet satisfy the
corresponding homogeneous equation; then find a particular solution
of the nonhomogeneous equation that does not involve any terms
from the homogeneous solution.
t²y" - t(6t+2)y' + (6t + 2)y = 5t³, t > 0.
NOTE: Use t as the independent variable.
Y(t) =
eTextbook and Media
Hint
Assistance Used
If the functions p, q, and g are continuous on an open interval I and if the functions y₁ and y₂ are a fundamental set of solutions of
the homogeneous equation y" +p(t)y + q(t)y = 0 corresponding to the nonhomogeneous equation y" +p(t)y + q(t)y = g(t).
then a particular solution of the nonhomogeneous equation is Y(1) = y₁ (1) 2(s)g(s) as + y₂ (1) /_y₁(3)g(s) as, where to is any
W(Y1-92)
To
To
W(₁32)(s)
conveniently chosen point in I. The general solution is y = C₁y₁ (1) + €₂y₂(1) + Y(t).
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