Paragraph: Suppose E is the differential equation representing family of curves y^2=2 c x+2 c c where c is a positive parameter. Question: What is the order of the differential equation?

Correct answer: A. 1 y^2=2 c x+2 c c Differentiate wrt x , 2 y (d y)/(d x)=2 c c=y (d y)/(d x) y^2=2 x · y (d y)/(d x)+2(y (d y)/(d x))^ 3 / 2 y^2-2 x y (d y)/(d x)=2 · (y (d y)/(d x))^ 3 / 2 Squaring both sides, y^4+4 x^2 y^2 ((d y)/(d x))^2-4 x y^3 (d y)/(d x)=4 y^3((d y)/(d x))^3 Clearly order =1 .

NDA 2023

Exam: NDA
Year: 2023
Subject: Mathematics
Chapter: Differential Equations

NDA 2023 · Mathematics · Differential Equations

Paragraph: Suppose E is the differential equation representing...

Paragraph: Suppose E is the differential equation representing family of curves

y^2=2 c x+2 c \sqrt{c}

where c is a positive parameter.
Question: What is the order of the differential equation?

Options

A $1$
B $2$
C $3$
D $4$

Correct answer

A. $1$

Step-by-step solution

y^2=2 c x+2 c \sqrt{c}

Differentiate wrt

x

2 y \frac{d y}{d x}=2 c\Rightarrow \quad c=y \frac{d y}{d x}\Rightarrow \quad y^2=2 x \cdot y \frac{d y}{d x}+2\left(y \frac{d y}{d x}\right)^{3 / 2}\Rightarrow y^2-2 x y \frac{d y}{d x}=2 \cdot

\left(y \frac{d y}{d x}\right)^{3 / 2}

Squaring both sides,

y^4+4 x^2 y^2

\left(\frac{d y}{d x}\right)^2-4 x y^3 \frac{d y}{d x}=4 y^3\left(\frac{d y}{d x}\right)^3

Clearly order

=1

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