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Chapter 13: Problem 1191

The solution of the difference equation \((\mathrm{dy} / \mathrm{dx})=\left[1 /\left\\{\mathrm{xy}\left(\mathrm{x}^{2} \sin \mathrm{y}^{2}+1\right)\right\\}\right]\) is: (A) \(x^{2}\left(\cos y^{2}-\sin y^{2}-e^{-(y) 2}\right)=4\) (B) \(y^{2}\left(\cos y^{2}-\sin y^{2}-2 c e^{-(y) 2}\right)=2\) (C) \(x^{2}\left(\cos y^{2}-\sin y^{2}-2 c e^{-(y) 2}\right)=2\) (D) none of these

Short Answer

Expert verified
The solution of the difference equation \(\frac{dy}{dx} = \frac{1}{xy(x^2\sin y^2 + 1)}\) is: (D) none of these

Step by step solution

01

Separate the variables

We need to express the given differential equation in the form \(g(y) dy = h(x) dx\). To do this, we can rewrite the given equation as: \(dy = \frac{1}{y(x^2\sin y^2 + 1)} dx\) Now, the variables are separated.
02

Integrate both sides

Next, we integrate both sides with respect to their respective variables: \(\int dy = \int \frac{1}{y(x^2\sin y^2 + 1)} dx\) On the left side, the integral is straightforward: \(\int dy = y + C_1\) On the right side, since the integral is not elementary, let's denote it by a function F(x): \(\int \frac{1}{y(x^2\sin y^2 + 1)} dx = F(x) + C_2\)
03

Combine the constants and express the solution

We can combine the constants on both sides by writing \(C = C_1 - C_2\), leading to the general solution: \(y + C = F(x)\) But this first-order equation cannot be expressed in elementary functions, which prevents us from solving it completely.
04

Compare the result with the provided options

Since we cannot integrate without any simplification or substitution, we can infer from this that none of the options match the form we have above. Therefore, the answer is: (D) none of these

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Most popular questions from this chapter

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

\(\underline{\text { Assertion - Reason Type Questions: }}\) 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.

Which of the following equations is a linear equation of order \(3 ?\) (A) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right) \cdot(\mathrm{dy} / \mathrm{dx})+\mathrm{y}=\mathrm{x}\) (B) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)+\mathrm{y}^{2}=\mathrm{x}^{2}\) (C) \(x \cdot\left(d^{3} y / d x^{3}\right)+\left(d^{3} y / d x^{3}\right)=e^{x}\) (D) \(\left(d^{2} y / d x^{2}\right)+(d y / d x)=\log x\)

The differential equation representing the family of curves \(\mathrm{y}^{2}=2 \mathrm{c}(\mathrm{x}+\mathrm{v} \mathrm{c})\), where \(\mathrm{c}\) is a positive parameter, is of order and degree as follows. (A) order 2, degree 1 (B) order 1 , degree 2 (C) order 2 , degree 2 (D) order 1 , degree 3

The particular solution of \(\left(1+\mathrm{y}^{2}\right) \mathrm{dx}+\left(\mathrm{x}-\mathrm{e}^{-(\tan )-1 \mathrm{y}}\right) \mathrm{dy}=0\) with initial condition \(\mathrm{y}(0)=0\) is: (A) \(x \cdot e^{(\tan )-1 x}=\cot ^{-1} x\) (B) \(x \cdot e^{(\tan )-1 y}=\tan ^{-1} y\) (C) \(x \cdot e^{(\tan )-1 y}=\cot ^{-1} y\) (D) \(x \cdot e^{(\cot )-1 y}=\tan ^{-1} y\)
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