Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
Question
If $=\text{y}=(\sin^{-1}\text{x})^2,$ prove that $(1-\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}-2=0.$
✓

Answer

Here $=\text{y}=(\sin^{-1}\text{x})^2$
$\Rightarrow\text{y}'=2(\sin^{-1}\text{x})\times\frac{1}{\sqrt{1-\text{x}^2}}$
$\Rightarrow\text{y}'\sqrt{1-\text{x}^2}=2(\sin^{-1}\text{x})$
$\Rightarrow\text{y}''\sqrt{1-\text{x}^2}+\text{y}'\times\frac{-2\text{x}}{2\sqrt{1-\text{x}^2}}=\frac{2}{\sqrt{1-\text{x}^2}}$
$\Rightarrow(1-\text{x}^2)\text{y}''-\text{xy}'=2$
$\therefore(1-\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}\frac{\text{dy}}{\text{dx}}-2=0.$

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

In order to supplement daily diet, a person wishes to take $X$ and $Y$ tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in $X$ and $Y$ are given as below:
Tablets Iron Calcium Vitamin
$X$ $6$ $3$ $2$
$Y$ $2$ $3$ $4$
The person needs to supplement at least $18$ milligrams of iron, $21$ milligrams of calcium and $16$ milligrams of vitamins. The price of each tablet of $X$ and $Y$ is $₹ 2$ and $₹1$ respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.
→
Find the maximum area of an isosceles $\triangle $ inscribed in the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ with its vertex at one end of the major axis
→
Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(–4, 4, 4).
→
Find the distance between the point (-1, -5, -10) and the point of intersection of the line $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{12}$ and the plane $x - y + z = 5.$
→
Solve the following systems of linear equations by cramer's rule:
5x - 7y + z = 11,
6x - 8y - z = 15,
3x + 2y - 6z = 7
→
Evaluate the following integrals:
$\int\limits^4_1\big\{|\text{x}-1|+|\text{x}-2|+|\text{x}-4\big\}\text{dx}$
→
Solve the following differential equation
$\text{x}\cos\text{y dy}=(\text{xe}^\text{x}\log\text{x}+\text{e}^\text{x})\text{dx}$
→
Prove that:
$\begin{vmatrix}\text{a}^2+1&\text{ab}&\text{ac}\\\text{ab}&\text{b}^2+1&\text{bc}\\\text{ca}&\text{cb}&\text{c}^2+1 \end{vmatrix}=1+\text{a}^2+\text{b}^2+\text{c}^2$
→
Using differentials, find the approximate values of the following:
$\sqrt{49.5}$
→
Evaluate the following intergrals:
$\int\text{e}^\text{ax}\sin(\text{bx}+\text{c})\text{dx}$
→