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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, then y can be obtained by integrating = v. Note that dy 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. Note: All solutions should be found. 8t²y" + (y')³ = 8ty', t > 0

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, then y can be obtained by integrating = v. Note that dy 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. Note: All solutions should be found. 8t²y" + (y')³ = 8ty', t > 0

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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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. Note: All solutions should be found. 8t²y" + (y')³ = 8ty', t> 0
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. Note: All solutions should be found. 8t²y" + (y')³ = 8ty', t> 0
NOTE: Use c₁ and c₂ as the constants of integration. y = ± , Y = C3
Transcribed Image Text:NOTE: Use c₁ and c₂ as the constants of integration. y = ± , Y = C3
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