Equations with the Dependent Variable Missing For a second order differential eq

Question

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 dy / dt = V. Note that 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

Explanation / Answer

c)

y = +- 4/3 (t-2C1)t+C1 + C2, y = C3

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