What is the differential equation whose solution represents the family c(y + c)^2 = x^3?

Category: QuestionsWhat is the differential equation whose solution represents the family c(y + c)^2 = x^3?
Editor">Editor Staff asked 11 months ago

What is the differential equation whose solution represents the family c(y + c)^2 = x^3?
 
(a) [2x/3 *(dy/dx) – y][x/3 *dy/dx]^2 = x^3
 
(b) [x/3 *(dy/dx) – y][x/3 *dy/dx]^2 = x^3
 
(c) [2x/3 *(dy/dx) – y][ 2x/3 *dy/dx]^2 = x^3
 
(d) [x/3 *(dy/dx) – y][ 2x/3 *dy/dx]^2 = x^3
 
This question was addressed to me during an interview for a job.
 
My question is taken from Linear First Order Differential Equations in portion Differential Equations of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

Correct choice is (c) [2x/3 *(dy/dx) – y][ 2x/3 *dy/dx]^2 = x^3
 
The best I can explain: The given family is c(y + c)^2 = x^3
 
Differentiating once, we get
 
c[2(y + c)]dy/dx = 3x^2
 
=> 2x^3 (y + c)/(y + c)^2 * dy/dx = 3x^2
 
=> 2x^3/(y + c) * dy/dx = 3x^2
 
Or, 2x (y + c)/(y + c)^2 * dy/dx = 3
 
=> 2x/3 *[dy/dx] = (y + c)
 
 => c = 2x/3 *[dy/dx] – y
 
Substituting c back into equation (1), we get
 
[2x/3 *(dy/dx) – y][ 2x/3 *dy/dx]^2 = x^3
 
which is the required differential equation