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Chapter 13: Problem 1151

The equation of a curve passing through \([2,(7 / 2)]\) and having gradient \(1-\left(1 / \mathrm{x}^{2}\right)\) at \((\mathrm{x}, \mathrm{y})\) is: (A) \(x y=x+1\) (B) \(x^{2}+x+1\) (C) \(x y=x^{2}+x+1\) (D) none of these

Short Answer

Expert verified
The correct answer is (D) none of these, as none of the given options match the obtained curve equation \[y(x) = x - \frac{1}{x} + 2\].

Step by step solution

01

Integrate the gradient equation

To obtain the equation of the curve from its gradient, we perform integration. We are given the gradient equation as: \[1- \frac{1}{x^2}\] Integrate this equation with respect to x: \[\int(1 - \frac{1}{x^2}) dx\]
02

Evaluate the integral

We have the integral as: \[\int(1 - \frac{1}{x^2}) dx = \int1 dx - \int\frac{1}{x^2} dx\] Now, integrate each term separately: Integral of first term is: \[\int1 dx = x + C_1\] Integral of second term is: \[\int\frac{1}{x^2} dx = -\frac{1}{x} + C_2\] Combining both integrals, we have: \[y(x) = x + C_1 - \frac{1}{x} + C_2\] Let's say, \(C = C_1 + C_2\) (Since, both are constants) So, the equation becomes: \[y(x) = x - \frac{1}{x} + C\]
03

Use the given point to determine the constant

We are given a point \( (2, \frac{7}{2})\) on the curve. Substitute these values into the equation: \[y(2) = 2 - \frac{1}{2} + C\] \[\frac{7}{2} = \frac{3}{2} + C\] Solve for C: \[C = \frac{7}{2} - \frac{3}{2} = 2\]
04

Write the final equation of the curve and choose the correct answer

With the obtained value of C, the final equation of the curve is: \[y(x) = x - \frac{1}{x} + 2\] Let's verify it with the given options: Option (A) is: \[x y = x + 1\] Multiplying both sides of our obtained equation by x, we get: \[x y = x^2 -1 + 2x\] This does not match option (A). Option (B) is an equation in x only, so it doesn't represent a curve with respect to x and y. Option (C) is: \[x y = x^2 + x + 1\] This does not match the equation we obtained. Thus, none of the given options match our obtained curve equation. Therefore, the correct answer is: (D) none of these

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Most popular questions from this chapter

If \(y=\left[x+\sqrt{ \left.\left(1+x^{2}\right)\right]^{n}}\right.\), then \(\left(1+x^{2}\right) \cdot\left(d^{2} y / d x^{2}\right)+x \cdot(d y / d x)\) - (A) - (B) \(2 \mathrm{x}^{2} \mathrm{y}\) (C) \(\mathrm{n}^{2} \mathrm{y}\) (D) \(-\mathrm{n}^{2} \mathrm{y}\)

The differential equation of all circles passing through the origin and having their centers on the \(\mathrm{x}\) -axis is: (A) \(y^{2}=x^{2}+2 x y(d y / d x)\) (B) \(y^{2}=x^{2}-2 x y(d y / d x)\) (C) \(x^{2}=y^{2}+x y(d y / d x)\) (D) \(x^{2}=y^{2}+3 x y(d y / d x)\)

Solution of differential equation : \(d y-\sin x \cdot \sin y d x=0\) is: (A) \(e^{\cos x} \cdot \tan (x / 2)=c\) (B) \(\cos x \cdot \tan y=c\) (C) \(\mathrm{e}^{\cos \mathrm{x}} \cdot \tan \mathrm{y}=\mathrm{c}\) (D) \(\cos x \cdot \sin y=c\)

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

The differential equation whose solution is \(\mathrm{Ax}^{2}+\mathrm{By}^{2}=1\), where \(\mathrm{A}\) and \(\mathrm{B}\) are arbitrary constants is of. (A) second order and second degree (B) first order and first degree (C) first order and second degree (D) second order and first degree
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