Question 1 (a) For x > 0, consider the homogeneous ordinary differential equation dy 1 (x² + y²) dx for the dependent variable y. (i) Introduce a new dependent variable z by z = and transform this homogeneous equation into a separable one for z. (ii) Obtain the general solution of this separable equation in the form G(x, z) = C for some function G and arbitrary constant C. (iii) Hence, obtain in the explicit form y = f(x) the unique solution of the homogeneous equation that satisfies y = 5 when x = 1.
Question 1 (a) For x > 0, consider the homogeneous ordinary differential equation dy 1 (x² + y²) dx for the dependent variable y. (i) Introduce a new dependent variable z by z = and transform this homogeneous equation into a separable one for z. (ii) Obtain the general solution of this separable equation in the form G(x, z) = C for some function G and arbitrary constant C. (iii) Hence, obtain in the explicit form y = f(x) the unique solution of the homogeneous equation that satisfies y = 5 when x = 1.
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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Can you help me with questions ii and iii please,
Thank you!
Transcribed Image Text:Question 1
(a)
For x > 0, consider the homogeneous ordinary differential equation
dy
1
(x² + y²)
2
dx
for the dependent variable y.
(i) Introduce a new dependent variable z by
Y
and transform this homogeneous equation into a separable one for z.
(ii) Obtain the general solution of this separable equation in the form
G(x, 2) = C
for some function G and arbitrary constant C.
(iii) Hence, obtain in the explicit form
y = f(x)
the unique solution of the homogeneous equation that satisfies y =
5 when x = 1.
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