SATHEE: Differential Equations 3 Question 2

Differential Equations 3 Question 2

2. The curve amongst the family of curves represented by the differential equation, $(x^{2} - y^{2}) d x + 2 x y d y = 0$ , which passes through $(1, 1)$ , is

(2019 Main, 10 Jan II)

(a) a circle with centre on the $Y$ -axis

(b) a circle with centre on the $X$ -axis

(d) a hyperbola with transverse axis along the $X$ -axis.

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Answer:

Correct Answer: 2. (b)

Solution:

Given differential equation is

$(x^{2} - y^{2}) d x + 2 x y d y = 0$ , which can be written as

$\frac{d y}{d x} = \frac{y^{2} - x^{2}}{2 x y}$

Put $y = v x [∵$ it is in homogeneous form $]$

$\Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$

Now, differential equation becomes

$\begin{aligned} v + x \frac{d v}{d x} = \frac{v^{2} x^{2} - x^{2}}{2 x (v x)} \Rightarrow v + x \frac{d v}{d x} = \frac{(v^{2} - 1) x^{2}}{2 v x^{2}} \\ \Rightarrow x \frac{d v}{d x} = \frac{v^{2} - 1}{2 v} - v = \frac{v^{2} - 1 - 2 v^{2}}{2 v} \\ \Rightarrow x \frac{d v}{d x} = - \frac{1 + v^{2}}{2 v} \Rightarrow \int \frac{2 v d v}{1 + v^{2}} = - \int \frac{d x}{x} \end{aligned}$

$\Rightarrow \ln (1 + v^{2}) = - \ln x - \ln C$

$\begin{array}{rrr} ∵ \int \frac{f^{'} (x)}{f (x)} d x \Rightarrow \ln | f (x) | + C \\ \Rightarrow \ln | (1 + v^{2}) C x | = 0 & [∵ \ln A + \ln B = \ln A B] \\ \Rightarrow (1 + v^{2}) C x = 1 & [\log_{e} x = 0 \Rightarrow x = e^{0} = 1] \end{array}$

Now, putting $v = \frac{y}{x}$ , we get

$1 + \frac{y^{2}}{x^{2}} C x = 1 \Rightarrow C (x^{2} + y^{2}) = x$

$∵$ The curve passes through $(1, 1)$ , so

$C (1 + 1) = 1 \Rightarrow C = \frac{1}{2}$

Thus, required curve is $x^{2} + y^{2} - 2 x = 0$ , which represent a circle having centre $(1, 0)$

$∴$ The solution of given differential equation represents a circle with centre on the $X$ -axis.

Differential Equations 3 Question 2

2. The curve amongst the family of curves represented by the differential equation, $(x^{2} - y^{2}) d x + 2 x y d y = 0$ , which passes through $(1, 1)$ , is

Answer:

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Differential Equations 3 Question 2

2. The curve amongst the family of curves represented by the differential equation, (x2−y2)dx+2xydy=0, which passes through (1,1), is

Answer:

Engineering Preparation

Medical Preparation

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Other Resources

Syllabus

Test Platform

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NCERT Exercise Video Solution

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2. The curve amongst the family of curves represented by the differential equation, $(x^{2} - y^{2}) d x + 2 x y d y = 0$ , which passes through $(1, 1)$ , is