A continuously differentiable function (x) , in ,(0, , ) satisfyi… - Vidyadip

MCQ

A continuously differentiable function $\phi (x)\,{\rm{in}}\,(0,\,\pi )$ satisfying $y' = 1 + {y^2},\,\,y(0) = 0 = y(\pi )$ is

A
$\tan x$
B
$x(x - \pi )$
C
$(x - \pi )$ $(1 - {e^x})$
✓
Not possible

✓

Answer

Correct option: D.

Not possible

d
(d) $\frac{{dy}}{{dx}} = 1 + {y^2}$ ==> $\frac{{dy}}{{1 + {y^2}}} = dx$

Integrating both sides,

$\int {\frac{{dy}}{{1 + {y^2}}} = \int {dx} } $ ==> ${\tan ^{ - 1}}y = x + c$

At $x = 0,$$y = 0,$ then $c = 0$

At $x = \pi ,$$y = 0,$ then ${\tan ^{ - 1}}0 = \pi + c$ ==> $c = - \pi $

$\therefore {\tan ^{ - 1}}y = x$==>$y = \tan x = \phi (x)$

Therefore, solution is $y = \tan x$

But $\tan x$ is not continuous function in $(0,\,\pi )$

Hence, $\phi \,(x)$ is not possible in $(0,\,\pi )$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from STD 12 - 9. differential equations→STD 12 - 9. differential equations — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

Let $S_n=\sum \limits_{k=1}^n k$, denotes the sum of the first $n$ positive integers. The numbers $S_1, S_2, S_3, \ldots, S_{99}$ are written on $99$ cards. The probability of drawing a card with an even number written on it is

The maximum value of $Z = 3x + 4y$ subjected to contraints $\text{x}+\text{y}\leq40,\text{x}+2\text{y}\leq60,\text{x}\geq0$ and $\text{y}\geq0$ is:

Let $f(x) = tan^{-1}\, (cot\, x - 2\, cot2x)$ then $\left[ {\sum\limits_{r = 1}^7 {f\left( r \right)} } \right]$ is equal to (where [.] represent greatest integer function)

If $\vec{\text{x}}$ is a vector in the direction of (2, -2, 1) of magnitude 6 and $\vec{\text{y}}$ is a vector in the direction of (1, 1, -1) of magnitude $\sqrt{3}$ then $\mid\vec{\text{x}}+2\vec{\text{y}}\mid=$

$\text { If } \int \frac{2 e^{x}+3 e^{-x}}{4 e^{x}+7 e^{-x}} d x=\frac{1}{14}\left(u x+v \log _{c}\left(4 e^{x}+7 e^{-x}\right)\right)+C$ where $\mathrm{C}$ is a constant of integration, then $\mathrm{u}+\mathrm{v}$ is equal to .... .

$\int_{}^{} {x{{\tan }^{ - 1}}} xdx = $

$\int_0^a {\frac{{{x^4}\,dx}}{{{{({a^2} + {x^2})}^4}}}} = $

Area of the region bounded by the curve $y^2 = 4x, y-$axis and the line $y = 3$, is:

Let $f : R \rightarrow R$ be a function defined by $\text{f(x)}=\frac{\text{x}^2-8}{\text{x}^2+2}.$ Then, $f$ is:

If $a = i + j + k,\,\,b = i + j,\,\,c = i$ and $(a \times b) \times c = \lambda \,a + \mu \,b$, then $\lambda + \mu = $