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, %3D 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. 50t²y" + (y')³ = 50ty', t > 0 NOTE: Use c1 and c2 as the constants of integration.
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, %3D 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. 50t²y" + (y')³ = 50ty', t > 0 NOTE: Use c1 and c2 as the constants of integration.
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: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
= v. Note that
dt
then y can be obtained by integrating
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.
50t?y" + (y) = 50ty', t > 0
3
NOTE: Use c, and cz as the constants of integration.
y = ±
,Y = C3
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