If the solution curve y = y(x) of the differential equation y2dx + (x2 – xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y = <math> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> <mi>x</mi> </mrow> </math> at the point <math> <mrow> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>,</mo> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math> then value of <math> <mrow> <mi>l</mi> <mi>o</mi> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> <mi>α</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> is equal to:

y 2 d x + ( x 2 − x y + y 2 ) d y = 0 d x d y + x 2 − x y + y 2 y 2 = 0 ⇒ d x d y + ( x y ) 2 − ( x y ) + 1 = 0 Put x = vy ⇒ v + y d v d y + v 2 − v + 1 = 0 ⇒ y d v d y + v 2 + 1 = 0 ⇒ π 6 + l n | y | = π 4 l n | y | = π 1 2

If the solution curve y = y(x) of the differential equation y2dx + (x2 – xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y = <math> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> <mi>x</mi> </mrow> </math> at the point <math> <mrow> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>,</mo> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math> then value of <math> <mrow> <mi>l</mi> <mi>o</mi> <msub> <mrow> <mi>g</mi> </mrow> <mrow> <mi>e</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> <mi>α</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> &nbsp;is equal to:

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Maths NCERT Exemplar Solutions Class 11th Chapter Two Maths Class 12th Differential Equations

If the solution curve y = y(x) of the differential equation y²dx + (x² – xy + y²)dy = 0, which passes through the point (1, 1) and intersects the line y = $\sqrt[]{3} x$ at the point $(α, \sqrt[]{3} α),$ then value of $l o g_{e} (\sqrt[]{3} α)$ is equal to:

Option 1 -

$\frac{π}{3}$

Option 2 -

$\frac{π}{2}$

Option 3 -

$\frac{π}{12}$

Option 4 -

$\frac{π}{6}$

2 Views | Posted 3 weeks ago

Asked by Shiksha User

1 Answer

Answered by

Raj Pandey | Contributor-Level 9

3 weeks ago

Correct Option - 3

Detailed Solution:

$y^{2} d x + (x^{2} - x y + y^{2}) d y = 0$

$\frac{d x}{d y} + \frac{x^{2} - x y + y^{2}}{y^{2}} = 0 \Rightarrow \frac{d x}{d y} + {(\frac{x}{y})}^{2} - (\frac{x}{y}) + 1 = 0$

Put x = vy $\Rightarrow v + y \frac{d v}{d y} + v^{2} - v + 1 = 0$

$\Rightarrow \frac{y d v}{d y} + v^{2} + 1 = 0$

$\Rightarrow \frac{π}{6} + l n | y | = \frac{π}{4}$

$l n | y | = \frac{π}{12}$

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If the solution curve y = y(x) of the differential equation y²dx + (x² – xy + y²)dy = 0, which passes through the point (1, 1) and intersects the line y = $\sqrt[]{3} x$ at the point $(α, \sqrt[]{3} α),$ then value of $l o g_{e} (\sqrt[]{3} α)$ is equal to:

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If the solution curve y = y(x) of the differential equation y2dx + (x2 – xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y = 3 x at the point ( α , 3 α ) , then value of l o g e ( 3 α ) is equal to:

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If the solution curve y = y(x) of the differential equation y²dx + (x² – xy + y²)dy = 0, which passes through the point (1, 1) and intersects the line y = $\sqrt[]{3} x$ at the point $(α, \sqrt[]{3} α),$ then value of $l o g_{e} (\sqrt[]{3} α)$ is equal to: