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Chapter 13: Problem 1147

Solution of \(\left(d^{2} y / d x^{2}\right)=\log x\) is: (A) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}-(3 / 4) \mathrm{x}^{2}+\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (B) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}+(3 / 4) \mathrm{x}^{2}+\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (C) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}-(3 / 4) \mathrm{x}^{2}-\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (D) None of these

Short Answer

Expert verified
The solution of the given differential equation is: \(y = \frac{1}{2}x^2 \log x - \frac{3}{4}x^2 + C_1x + C_2\).

Step by step solution

01

Integrate the differential equation once

Let's integrate the given differential equation with respect to x once: \[\int \frac{d^2 y}{dx^2} dx = \int \log x dx\] Let the first derivative of y, \(\frac{dy}{dx}\), be denoted by \(v\). Then we have: \[\frac{dy}{dx} = v = \int \log x dx + C_1\]
02

Integrate the differential equation again

Now we need to integrate the equation we derived above: \[\int \frac{dy}{dx} dx = \int v dx = \int \left( \int \log x dx + C_1 \right) dx\] Using integration by parts, we can find the value of \(\int \log x dx\): Let u = \(\log x\) and \(dv = dx\), then: \(du = \frac{1}{x} dx\) and \(v = x\). Now, applying integration by parts: \[\int \log x dx = x \log x - \int x (\frac{1}{x}dx)\] \[\int \log x dx = x \log x - \int dx\] \[\int \log x dx = x \log x - x + C\] Now, substitute this value back into the equation: \[y = \int (x \log x - x + C + C_1) dx\] \[y = \frac{1}{2}x^2 \log x - \frac{1}{2}x^2 + Cx + C_1x + C_2\] Comparing this with the given options:
03

Compare the derived equation with given options

The derived equation matches option (A): \(y = \frac{1}{2}x^2 \log x - \frac{3}{4}x^2 + C_1x + C_2\) So the correct solution is Option (A).

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Most popular questions from this chapter

Family of curves \(\mathrm{y}=\mathrm{e}^{\mathrm{x}}(\mathrm{A} \cos \mathrm{x}+\mathrm{B} \sin \mathrm{x})\) represents the differential equation: \(\quad\) (where \(\mathrm{A}\) and \(\mathrm{B}\) are arbitrary constant) (A) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)-2(\mathrm{dy} / \mathrm{dx})+\mathrm{y}=0\) (B) \(\left(d^{2} y / d x^{2}\right)-2(d y / d x)-2 y=0\) (C) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)-2(\mathrm{dy} / \mathrm{dx})-\mathrm{y}=0\) (D) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right)-2(\mathrm{dy} / \mathrm{dx})+2 \mathrm{y}=0\)

\(\underline{\text { Assertion - Reason Type Questions: }}\) Each question has four choices (a), (b), (c) and (d) out of which only one is correct. Write (a), (b), (c) and (d) according to the following rules. (a) Statement- 1 is True, Statement-2 is True, Statement- 2 is a correct explanation for Statement-1. (b) Statement-1 is True, Statement- 2 is True, Statement-2 is not a correct explanation for Statement-1. (c) Statement- 1 is True, Statement- 2 is False. (d) Statement- 1 is False, Statement- 2 is True. Statement \(-2:\) The differential equation \(\mathrm{y}^{\prime}=(\mathrm{y} / 2 \mathrm{x})\) is variable separable. Statement-1: Curve satisfying the differential equation \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / 2 \mathrm{x})\) passing through \((2,1)\) is a parabola with Focus \([(1 / 4), \underline{0}]\). Statement- 2 : The differential equation \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / 2 \mathrm{x})\) is variable separable.

The solution of the equation \(\mathrm{x}+\mathrm{y}(\mathrm{dy} / \mathrm{d} \mathrm{x})=2 \mathrm{y}\) is: (A) \(x y^{2}=c^{2}(x+2 y)\) (B) \(\mathrm{y}^{2}=\mathrm{c}\left(\mathrm{x}^{2}+2 \mathrm{y}\right)\) (C) \(\log (\mathrm{y}-\mathrm{x})=\mathrm{c}+[\mathrm{x} /(\mathrm{y}-\mathrm{x})]\) (D) \(\log [\mathrm{x} /(\mathrm{x}-\mathrm{y})]=\mathrm{c}+\mathrm{y}-\mathrm{x}\)

Differential equation of the curves having the subnormal with \((7 / 2)\) units and passes through \((0,0)\) is: (A) \(x^{2}=7 y\) (B) \(\mathrm{y}^{2}=7 \mathrm{x}+\mathrm{c}, \mathrm{c} \neq 0\) (C) \(y^{2}=7 x\) (D) None of these

The degree and order of the differential equation of the family of all parabolas whose axis is \(\mathrm{x}\) -axis, are respectively. (A) 1,2 (B) 3,2 (C) 2,3 (D) 2,1
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