3. The differential equation dy != y − 4t + y² − 8yt + 16t² +4. dt is not autonomous, separable, or linear; however, we can solve this equa- tion with a change of variable. (a) Transform this equation into a new differential equation of the form du dt = f(u) by letting u = y - 4t. (b) Sketch the phase line for this new equation, u' = f(u), and sketch several solutions. (e) Find the solutions of the original differential equation that corre- spond to the equilibrium solutions of u' = f(u). Graph these solu- tions in ty-plane. Also, sketch the graphs of the solutions that you plotted in part (b). (d) Solve the differential new equation and use this information to solve the original differential equation.
3. The differential equation dy != y − 4t + y² − 8yt + 16t² +4. dt is not autonomous, separable, or linear; however, we can solve this equa- tion with a change of variable. (a) Transform this equation into a new differential equation of the form du dt = f(u) by letting u = y - 4t. (b) Sketch the phase line for this new equation, u' = f(u), and sketch several solutions. (e) Find the solutions of the original differential equation that corre- spond to the equilibrium solutions of u' = f(u). Graph these solu- tions in ty-plane. Also, sketch the graphs of the solutions that you plotted in part (b). (d) Solve the differential new equation and use this information to solve the original differential equation.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Confused on parts a through d, needing some assistance.
Transcribed Image Text:3.
The differential equation
dy
!= y − 4t + y² − 8yt + 16t² +4.
dt
is not autonomous, separable, or linear; however, we can solve this equa-
tion with a change of variable.
(a) Transform this equation into a new differential equation of the form
du
dt
= f(u)
by letting u = y - 4t.
(b) Sketch the phase line for this new equation, u' = f(u), and sketch
several solutions.
(e) Find the solutions of the original differential equation that corre-
spond to the equilibrium solutions of u' = f(u). Graph these solu-
tions in ty-plane. Also, sketch the graphs of the solutions that you
plotted in part (b).
(d) Solve the differential new equation and use this information to solve
the original differential equation.
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