Question:
The slope of a curve at any point \((x,y)\), other than origin, is \( y + \frac{y}{x} \). Then the equation of the curve is
The slope of a curve at any point \((x,y)\), other than origin, is \( y + \frac{y}{x} \). Then the equation of the curve is
Show Hint
Convert into separable form whenever possible—it simplifies solving differential equations quickly.
Updated On: May 8, 2026
- \( y = Cxe^x \)
- \( y = x(e^x + C) \)
- \( xy = Ce^x \)
- \( y + xe^x = C \)
- \( (y-x)e^x = C \)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Concept:
Given slope:
\[
\frac{dy}{dx} = y + \frac{y}{x}
\]
This is a first-order linear differential equation
Step 1: Rewrite equation
\[ \frac{dy}{dx} = y\left(1 + \frac{1}{x}\right) \]
Step 2: Separate variables
\[ \frac{1}{y} dy = \left(1 + \frac{1}{x}\right) dx \]
Step 3: Integrate both sides
\[ \int \frac{1}{y}dy = \int \left(1 + \frac{1}{x}\right)dx \] \[ \ln|y| = x + \ln|x| + C \]
Step 4: Simplify
\[ \ln|y| - \ln|x| = x + C \] \[ \ln\left|\frac{y}{x}\right| = x + C \]
Step 5: Exponentiate
\[ \frac{y}{x} = Ce^x \] \[ y = xCe^x \] \[ xy = Ce^x \] Final Answer: \[ \boxed{xy = Ce^x} \]
Step 1: Rewrite equation
\[ \frac{dy}{dx} = y\left(1 + \frac{1}{x}\right) \]
Step 2: Separate variables
\[ \frac{1}{y} dy = \left(1 + \frac{1}{x}\right) dx \]
Step 3: Integrate both sides
\[ \int \frac{1}{y}dy = \int \left(1 + \frac{1}{x}\right)dx \] \[ \ln|y| = x + \ln|x| + C \]
Step 4: Simplify
\[ \ln|y| - \ln|x| = x + C \] \[ \ln\left|\frac{y}{x}\right| = x + C \]
Step 5: Exponentiate
\[ \frac{y}{x} = Ce^x \] \[ y = xCe^x \] \[ xy = Ce^x \] Final Answer: \[ \boxed{xy = Ce^x} \]
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Differential equations Questions
- An integrating factor of the differential equation $xdy - ydx + x^2e^xdx = 0$ is
- If $xy\, = \,A \,sinx \,+ \,B \,cos \,x$ is the solution of the differential equation $x\frac{d^{2}y}{dx^{2}}-5a\frac{dy}{dx}+xy=0$ then the value of $a$ is equal to
- The order of the differential equation $\left(\frac{d^{3}\, y }{dx^{3}}\right)^{2} + \left(\frac{d^{2}\,y}{dx}\right)^{2} + \left(\frac{dy}{dx}\right)^{5} = 0 $ is
- The solution of $\frac{dy}{dx} + y \, \tan \, x = \sec \, x, y (0) = 0$ is
- If $ {{c}_{1}},{{c}_{2}},{{c}_{3}},{{c}_{4}},{{c}_{5}} $ and $ {{c}_{6}} $ are constants, then the order of the differential equation whose general solution is given by $ y={{c}_{1}}cos $ $ (x+{{c}_{2}})+{{c}_{3}}\sin (x+{{c}_{4}})+{{c}_{5}}{{e}^{x}}+{{c}_{6}} $
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions