SATHEE: Differential Equations 2 Question 8

Differential Equations 2 Question 8

8. If a curve passes through the point $(1, - 2)$ and has slope of the tangent at any point $(x, y)$ on it as $\frac{x^{2} - 2 y}{x}$ , then the curve also passes through the point

(a) $(\sqrt{3}, 0)$

(b) $(- 1, 2)$

(d) $(3, 0)$

(2019 Main, 12 Jan II)

Show Answer

Answer:

Correct Answer: 8. (a)

Solution:

We know that, slope of the tangent at any point $(x, y)$ on the curve is

$\begin{aligned} \frac{d y}{d x} & = \frac{x^{2} - 2 y}{x} \\ \Rightarrow \frac{d y}{d x} + \frac{2}{x} y & = x \end{aligned}$

which is a linear differential equation of the form $\frac{d y}{d x} + P (x) \cdot y = Q (x)$

where

$P (x) = \frac{2}{x} and Q (x) = x$

Now, integrating factor

$\begin{aligned} (IF) & = e^{\int P (x) d x} = e^{\int \frac{2}{x} d x} = e^{2 \log_{e} x} \\ = e^{\log_{e} x^{2}} & [∵ m \log a = \log a^{m}] \\ = x^{2} & [∵ e^{\log_{e} f (x)} = f (x)] \end{aligned}$

and the solution of differential Eq. (i) is

$\begin{array}{cc} y (I F) = \int Q (x) (I F) d x + C \Rightarrow y (x^{2}) = \int x \cdot x^{2} d x + C \\ \Rightarrow & y x^{2} = \frac{x^{4}}{4} + C \end{array}$

$∵$ The curve (ii) passes through the point $(1, - 2)$ , therefore

$- 2 = \frac{1}{4} + C \Rightarrow C = - \frac{9}{4}$

$∴$ Equation of required curve is $4 y x^{2} = x^{4} - 9$ .

Now, checking all the option, we get only $(\sqrt{3}, 0)$ satisfy the above equation.

Differential Equations 2 Question 8

8. If a curve passes through the point $(1, - 2)$ and has slope of the tangent at any point $(x, y)$ on it as $\frac{x^{2} - 2 y}{x}$ , then the curve also passes through the point

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Differential Equations 2 Question 8

8. If a curve passes through the point (1,−2) and has slope of the tangent at any point (x,y) on it as x2−2yx, then the curve also passes through the point

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

8. If a curve passes through the point $(1, - 2)$ and has slope of the tangent at any point $(x, y)$ on it as $\frac{x^{2} - 2 y}{x}$ , then the curve also passes through the point