If x = int_{-y}^{y} frac{dt}{sqrt{1 + 9t^2}} | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given $x = \int_{-y}^{y} \frac{dt}{\sqrt{1 + 9t^2}}$.Since the integrand $f(t) = \frac{1}{\sqrt{1 + 9t^2}}$ is an even function,we can write $x =

Vedclass · Accepted Answer

$\frac{9}{4}$

Vedclass · Answer

(B) Given $x = \int_{-y}^{y} \frac{dt}{\sqrt{1 + 9t^2}}$.Since the integrand $f(t) = \frac{1}{\sqrt{1 + 9t^2}}$ is an even function,we can write $x = 2 \int_{0}^{y} \frac{dt}{\sqrt{1 + 9t^2}}$.Differentiating both sides with respect to $y$ using the Fundamental Theorem of Calculus,we get $\frac{dx}{dy} = 2 \cdot \frac{1}{\sqrt{1 + 9y^2}}$.Taking the reciprocal,$\frac{dy}{dx} = \frac{\sqrt{1 + 9y^2}}{2} = \frac{1}{2} (1 + 9y^2)^{1/2}$.Now,differentiate with respect to $x$ to find the second derivative:$\frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{1}{2} (1 + 9y^2)^{1/2} \right) = \frac{1}{2} \cdot \frac{1}{2} (1 + 9y^2)^{-1/2} \cdot (18y) \cdot \frac{dy}{dx}$.Substitute $\frac{dy}{dx} = \frac{\sqrt{1 + 9y^2}}{2}$ into the equation:$\frac{d^2y}{dx^2} = \frac{1}{4} \cdot \frac{18y}{\sqrt{1 + 9y^2}} \cdot \frac{\sqrt{1 + 9y^2}}{2} = \frac{18y}{8} = \frac{9}{4}y$.Comparing this with $\frac{d^2y}{dx^2} = ky$,we get $k = \frac{9}{4}$.

If $x = \int_{-y}^{y} \frac{dt}{\sqrt{1 + 9t^2}}$ and $\frac{d^2y}{dx^2} = ky$,then $k$ equals

Similar Questions

The differential equation $\frac{dx}{dy} = \frac{3y}{2x}$ represents a family of hyperbolas (except when it represents a pair of lines) with eccentricity:

If $y = \frac{x}{\log_e|cx|}$ is the solution of the differential equation $\frac{dy}{dx} = \frac{y}{x} + \phi\left(\frac{x}{y}\right)$,then $\phi\left(\frac{x}{y}\right)$ is given by

The solution of $\frac{d^2y}{dx^2} = \sec^2 x + x e^x$ is

The solution of the differential equation $\cos^2 x \frac{d^2y}{dx^2} = 1$ is

Let $f:(-1,1) \rightarrow R$ be a differentiable function satisfying $(f^{\prime}(x))^4 = 16(f(x))^2$ for all $x \in (-1,1)$ and $f(0)=0$. The number of such functions is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module