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Chapter 13: Problem 1184

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

Short Answer

Expert verified
The short answer to the problem is: (D) None of these The differential equation of the curves having a subnormal of \((7 / 2)\) units and passing through the point \((0, 0)\) is given by the equation: \(x^2 = e^{\frac{4y}{7}} + C\) None of the given options match the derived equation.

Step by step solution

01

Noting the Given Information

We are given: 1. The length of the subnormal is \((7 / 2)\) units. 2. The curves pass through the point \((0, 0)\).
02

Write the Expression for Subnormal Length

Let \(y = f(x)\) be the curve. Its differential is: \[ dy = f'(x) dx \] If \(P(x, y)\) is any point on the curve and its subnormal length is \(p\), we have: \[ p = x \tan \theta \] where \(\theta\) is the angle between the tangent at \(P\) and the x-axis. Note that \(\tan \theta = \frac{dy}{dx} = f'(x)\). From the given information, we have \(p = \frac{7}{2}\).
03

Write the Expression for the Derivative and Integrate

By substituting the value of \(p\) in the equation for subnormal, we get: \[ x f'(x) = \frac{7}{2} \] Now, we need to find \(f(x)\) by integrating with respect to \(x\). So, we rewrite the equation as: \[ f'(x) = \frac{7}{2x} \] Now integrate on both sides: \[ \int f'(x) dx = \int \frac{7}{2x} dx \] Using integration, we get: \[ f(x) = \frac{7}{2} \ln |x| + C \] Here, \(C\) is an arbitrary constant.
04

Applying the Given Condition (\(0, 0\))

The curve passes through the point \((0, 0)\). So, we can use \((x, y) = (0, 0)\) to find the value of \(C\). However, since ln(x) is undefined at x = 0, we must use L'Hôpital's rule to find the limit as x approaches 0 for f(x). Consider the limit: \[ \lim_{x \to 0} \frac{y}{\ln x} = \lim_{x \to 0} \frac{f(x)}{\ln x} \] Applying L'Hôpital's rule: \[ \lim_{x \to 0} \frac{f'(x)}{1/x} = \lim_{x \to 0} \frac{7/2x}{1/x} \] Evaluating the limit, we get: \[ \lim_{x \to 0} \frac{7/2x}{1/x} = \frac{7}{2} \] So, the equation of the curve becomes: \[ y = \frac{7}{2} \ln |x| \]
05

Rewrite Equation in Implicit Form using Exponentials

Finally, we use exponentials to rewrite the equation in an implicit form: First, let \(2y = 7 \ln |x|\), then: \[ \frac{2y}{7} = \ln |x| \] Now, using exponentials: \[ |x| = e^{\frac{2y}{7}} \] In implicit form, we have: \[ x^2 = e^{\frac{4y}{7}} + C \] Now, we have to compare our derived equation with the given options: (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 Since no given options match our derived equation, the correct answer is (D) None of these.

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Most popular questions from this chapter

\(\mathrm{y}^{2}=(\mathrm{x}-\mathrm{c})^{3}\) is general solution of the differential equation: (where c is arbitrary constant). (A) \((\mathrm{dy} / \mathrm{dx})^{3}=27 \mathrm{y}\) (B) \(2(\mathrm{dy} / \mathrm{dx})^{3}-8 \mathrm{y}=0\) (C) \(8(\mathrm{dy} / \mathrm{dx})^{3}=27 \mathrm{y}\) (D) \(8\left(\mathrm{~d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)-27 \mathrm{y}=0\)

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) \(\cos x \cdot \sin y=c\)

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) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right)-2(\mathrm{dy} / \mathrm{dx})+2 \mathrm{y}=0\)

The solution of the differential equation \(x^{2}(d y / d x)-x y=1+\cos (y / x)\) is: (A) \(\tan (\mathrm{y} / \mathrm{x})=\mathrm{c}+(1 / \mathrm{x})\) (B) \(\tan (\mathrm{y} / 2 \mathrm{x})=\mathrm{c}-\left[1 /\left(2 \mathrm{x}^{2}\right)\right]\) (C) \(\cos (\mathrm{y} / \mathrm{x})=1+(\mathrm{c} / \mathrm{x})\) (D) \(x^{2}=\left(c+x^{2}\right) \cdot \tan (\mathrm{y} / \mathrm{x})\)

The differential equation of family of parabolas with focus at origin and \(\mathrm{x}\) -axis as axis is: (A) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}-2 \mathrm{x}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\) (B) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}+2 \mathrm{xy}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\) (C) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}-2 \mathrm{xy}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\) (D) \(\mathrm{y}(\mathrm{dy} / \mathrm{dx})^{2}+2 \mathrm{x}(\mathrm{dy} / \mathrm{dx})=\mathrm{y}\)
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