Equations with the Dependent Variable Missing For a second-order differential equation of the form y" = f(t, y'), the substitution v = y', v' = =y" leads to a first-order equation of the form v' = f(t, v). If this equation can be solved for v, dy then y can be obtained by integrating = v. Note that dt one arbitrary constant is obtained in solving the first-order equation for v, and a second is introduced in the integration for y. Use this substitution to solve the given equation. (1+t²)y" + 2ty'+33t² = 0, y(1) = 22, y'(1) = −11

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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y(t) =
Transcribed Image Text:y(t) =
Equations with the Dependent Variable Missing
For a second-order differential equation of the form y" = f(t, y'),
the substitution v = y', v' = =y" leads to a first-order equation
of the form v' = f(t, v). If this equation can be solved for v,
dy
then y can be obtained by integrating = v. Note that
dt
one arbitrary constant is obtained in solving the first-order
equation for v, and a second is introduced in the integration
for
y. Use this substitution to solve the given equation.
(1+t²)y" + 2ty'+33t² = 0, y(1) = 22, y'(1) = −11
Transcribed Image Text:Equations with the Dependent Variable Missing For a second-order differential equation of the form y" = f(t, y'), the substitution v = y', v' = =y" leads to a first-order equation of the form v' = f(t, v). If this equation can be solved for v, dy then y can be obtained by integrating = v. Note that dt one arbitrary constant is obtained in solving the first-order equation for v, and a second is introduced in the integration for y. Use this substitution to solve the given equation. (1+t²)y" + 2ty'+33t² = 0, y(1) = 22, y'(1) = −11
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