Q108 / 180 · COMEDK 2019 · Mathematics

The differential equation $y \frac{d y}{d x}+x=c$ represents

  • A
    a family of hyperbolas.
  • B
    a family of circles whose centres are on the $Y$-axis.
  • C
    a family of parabolas.
  • Correct answer
    D
    a family of circles whose centres are on the $X$-axis.

Solution

We have, $$ \begin{gathered} y \frac{d y}{d x}+x=c \\ \Rightarrow \quad y \frac{d y}{d x}=(c-x) \\ \Rightarrow \quad y d y=(c-x) d x \end{gathered} $$ $$ \begin{array}{ll} \Rightarrow & y d y=-(x-c) d x \\ \Rightarrow & \int y d y=-\int(x-c) d x \\ \Rightarrow \quad & \frac{y^{2}}{2}=-\frac{(x-c)^{2}}{2}+K \\ \Rightarrow & (x-c)^{2}+y^{2}=2 K \end{array} $$ which is a circle with centre $(c, 0)$ on the $X$-axis.

Hence, the correct answer is option: 'D'.

Relevant Questions