Limit Evaluation Steps
The given limit is:
$ \lim_{t \to x} \left( \frac{t^2 y(x) - x^2 y(t)}{x - t} \right) = 3 $This limit takes the indeterminate form $0/0$ as $t$ approaches $x$. We apply L'Hôpital's Rule, differentiating the numerator and denominator with respect to $t$:
- Numerator derivative: $\frac{d}{dt} (t^2 y(x) - x^2 y(t)) = 2t y(x) - x^2 y'(t)$
- Denominator derivative: $\frac{d}{dt} (x - t) = -1$
Applying the rule and evaluating the limit at $t = x$:
$ \frac{2x y(x) - x^2 y'(x)}{-1} = x^2 y'(x) - 2x y(x) $Equating this expression to 3 yields the differential equation:
$ x^2 y'(x) - 2x y(x) = 3 $Differential Equation Solution
Rewrite the equation in standard linear form:
$ y'(x) - \frac{2}{x} y(x) = \frac{3}{x^2} $This is a first-order linear ordinary differential equation (ODE). Calculate the integrating factor $I(x)$:
$ I(x) = e^{\int -\frac{2}{x} dx} = e^{-2 \ln|x|} $Since the domain is $(0, \infty)$, $|x| = x$:
$ I(x) = e^{-2 \ln x} = e^{\ln(x^{-2})} = x^{-2} = \frac{1}{x^2} $Multiply the ODE by the integrating factor $I(x)$:
$ \frac{1}{x^2} y'(x) - \frac{2}{x^3} y(x) = \frac{3}{x^4} $The left side is the derivative of the product $I(x)y(x)$:
$ \frac{d}{dx} \left( \frac{1}{x^2} y(x) \right) = \frac{3}{x^4} $Integrate both sides with respect to $x$:
$ \frac{1}{x^2} y(x) = \int \frac{3}{x^4} dx = 3 \int x^{-4} dx $ $ \frac{1}{x^2} y(x) = 3 \left( \frac{x^{-3}}{-3} \right) + C = -x^{-3} + C $Solve for the general solution $y(x)$:
$ y(x) = x^2 \left( -\frac{1}{x^3} + C \right) = -\frac{1}{x} + C x^2 $Initial Condition Application
Use the given initial condition $y(1) = 2$ to find the constant $C$:
$ 2 = -\frac{1}{1} + C (1)^2 $ $ 2 = -1 + C \implies C = 3 $Substitute $C = 3$ back into the general solution to get the specific solution:
$ y(x) = -\frac{1}{x} + 3x^2 $Final Value Calculation
Evaluate the function at $x = 2$:
$ y(2) = -\frac{1}{2} + 3(2)^2 = -\frac{1}{2} + 3(4) = -\frac{1}{2} + 12 $ $ y(2) = \frac{-1 + 24}{2} = \frac{23}{2} $Calculate the required value $2y(2)$:
$ 2y(2) = 2 \times \frac{23}{2} = 23 $Similar Questions
Let $[\cdot]$ be the greatest integer function. If $\alpha = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) dx$, then $\frac{1}{\pi} \int_{0}^{\alpha \pi} \left(\frac{\sin^2 \theta}{\sin^6 \theta + \cos^6 \theta}\right) d\theta$ is equal to ________.
- If $f(x)$ satisfies the relation $f(x) = e^x + \int_0^1 (y + xe^x)f(y)dy$, then $e + f(0)$ is equal to _______.
- $6 \int_0^\pi |(\sin 3x + \sin 2x + \sin x)| dx$ is equal to ____________.
- Let $y = y(x)$ be the solution of the differential equation $\sec x \frac{dy}{dx} - 2y = 2 + 3\sin x$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, $y(0) = -\frac{7}{4}$. Then $y\left(\frac{\pi}{6}\right)$ is equal to :
- If $\int_0^1 4 \cot^{-1}(1 - 2x + 4x^2) dx = a \tan^{-1}(2) - b \log_e(5)$, where $a, b \in \mathbf{N}$, then $(2a + b)$ is equal to __________.
- Let the area of the region bounded by the curve $y = \max\{\sin x, \cos x\}$, lines $x = 0$, $x = \frac{3\pi}{2}$, and the $x$-axis be $A$. Then, $A + A^2$ is equal to ___________.
- The number of elements in the set $S = \left\{ x : x \in [0, 100] \text{ and } \int_{0}^{x} t^2 \sin(x-t)dt = x^2 \right\}$ is _____
- If the solution curve $y = f(x)$ of the differential equation
$(x^2-4)y' - 2xy + 2x(4-x^2)^2 = 0$, $x > 2$,
passes through the point (3, 15), then the local maximum value of $f$ is _____ - Let $y = y(x)$ be the solution of the differential equation $x \sin\left(\frac{y}{x}\right) dy = \left(y \sin\left(\frac{y}{x}\right) - x\right) dx$, $y(1) = \frac{\pi}{2}$ and let $\alpha = \cos\left(\frac{y(e^{12})}{e^{12}}\right)$. Then the number of integral value of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$, is _________.
Let $y=y(x)$ be the solution of the differential equation $(\tan x)^{\frac{1}{2}} \text{ dy} = \left(\sec^{3}x - (\tan x)^{\frac{3}{2}}y\right) \text{ dx}, \ 0 < x < \frac{\pi}{2}, \ y\left(\frac{\pi}{4}\right) = \frac{6\sqrt{2}}{5}$. If $y\left(\frac{\pi}{3}\right) = \frac{4}{5} \alpha$, then $\alpha^{4}$ equals ___________.
Important Questions from Calculus
Let $[\cdot]$ be the greatest integer function. If $\alpha = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) dx$, then $\frac{1}{\pi} \int_{0}^{\alpha \pi} \left(\frac{\sin^2 \theta}{\sin^6 \theta + \cos^6 \theta}\right) d\theta$ is equal to ________.
- If $f(x)$ satisfies the relation $f(x) = e^x + \int_0^1 (y + xe^x)f(y)dy$, then $e + f(0)$ is equal to _______.
- $6 \int_0^\pi |(\sin 3x + \sin 2x + \sin x)| dx$ is equal to ____________.
- Let $y = y(x)$ be the solution of the differential equation $\sec x \frac{dy}{dx} - 2y = 2 + 3\sin x$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, $y(0) = -\frac{7}{4}$. Then $y\left(\frac{\pi}{6}\right)$ is equal to :
- If $\int_0^1 4 \cot^{-1}(1 - 2x + 4x^2) dx = a \tan^{-1}(2) - b \log_e(5)$, where $a, b \in \mathbf{N}$, then $(2a + b)$ is equal to __________.
