ધારોકે y=y(x) એ વિકલ સમીકરણ (1-x^ 2 ) d y= (x y+ (x^ 3 +2 ) 1-x^ … - Vidyadip

MCQ

ધારોકે $y=y(x)$ એ વિકલ સમીકરણ $\left(1-x^{2}\right) d y=\left(x y+\left(x^{3}+2\right) \sqrt{1-x^{2}}\right) d x,-1< x < 1$ અને $y(0)=0$ જો $\int\limits_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^{2}} y(x) d x=k$ હોય તો $k^{-1}$ ની કિમંત મેળવો.

A
$320$
B
$321$
✓
$322$
D
$323$

✓

Answer

Correct option: C.

$322$

c
$\left(1-x^{2}\right) \frac{d y}{d x}=x y+\left(x^{3}+2\right) \sqrt{1-x^{2}}$

$\Rightarrow \frac{d y}{d x}+\left(\frac{-x}{1-x^{2}}\right) y=\frac{x^{3}+2}{\sqrt{1-x^{2}}}$

$I F=e^{\int \frac{-x}{1-x^{2}} d x}=\sqrt{1-x^{2}}$

$y(x) \cdot \sqrt{1-x^{2}}=\frac{x^{4}}{4}+2 x+c$

$y (0)=0 \Rightarrow c =0$

$\sqrt{1-x^{2}} y(x)=\frac{x^{4}}{4}+2 x$

required value $=\int\limits_{-1 / 2}^{1 / 2}\left(\frac{x^{4}}{4}+2 x\right) d x-\frac{1}{4} \cdot 2 \int\limits_{0}^{1 / 2} x^{4} d x$

$=\frac{1}{10}\left(x^{5}\right)_{0}^{1 / 2}=\frac{1}{320}$

$k ^{-1}=320$

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 "MCQ" from 9. differential equations→9. differential equations — all question types→ગણિત ધોરણ 12 સાયન્સ question papers→Gujarat Board ધોરણ 12 સાયન્સ question papers→Gujarat Board question paper generator→

Similar questions

મુખ્ય કિંમત શોધો : $\cos ^{-1}\left(-\frac{1}{\sqrt{2}}\right)$

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\{b + c}&{c + a}&{a + b}\\{b + c - a}&{c + a - b}&{a + b - c}\end{array}\,} \right|$ = . .

Ravi and Rashmi are each holding $2$ red cards and $2$ black cards (all four red and all four black cards are identical). Ravi picks a card at random from Rashmi and then Rashmi picks a card to random from Ravi. This process is repeated a second time. Let $p$ be the probability that both have all $4$ cards of the same colour. Then, $p$ satisfies

જો $\mathrm{U}_{\mathrm{n}}=\left(1+\frac{1}{\mathrm{n}^{2}}\right)\left(1+\frac{2^{2}}{\mathrm{n}^{2}}\right)^{2} \ldots\left(1+\frac{\mathrm{n}^{2}}{\mathrm{n}^{2}}\right)^{\mathrm{n}}$, હોય તો $\lim _{n \rightarrow \infty}\left(U_{n}\right)^{\frac{-4}{n^{2}}}$ ની કિમંત મેળવો.

વિધેય $f(x) = {x^2} - 4$ એ . . . . અંતરાલમાં રોલના પ્રમેય નું પાલન કરે છે .

$f(x) = \left| {\left| x \right| - 1} \right|$ એ . . . . આગળ વિકલનીય નથી.

જો $S = \{\lambda ,\mu \} \in R \times R:f\left( t \right) = \left( {\left| \lambda \right|{e^{\left| t \right|}} - \mu } \right)$. $\sin \left( {2\left| t \right|} \right),t \in R$ , એ વિકલનીય વિધેય છે $\}$ . તો $S$ એ કોનો ઉપગણ બને ?

$\int_{-2}^{0}[x^3+3x^2+3x+1+(x+1)\cos(x+1)]dx =\ ..........$

જ્યારે $r=7$ સેમી હોય ત્યારે ગોલકના ધનફળનો ત્રિજ્યાને સાપેક્ષ બદલવાનો દર........... છે.

વિધેય $f\left( x \right) = \left( {\frac{{{e^{2x}} - 1}}{{{e^{2x}} + 1}}} \right)$ એ કેવું વિધેય છે $?$