KCET · Maths · Differential Equations
The differential equation \(y \frac{d y}{d x}+x=\) crepresents
- A a family of hyperbolas
- B a family of circles whose centres are on the axis
- C a family of parabolas
- D a family of circles whose centres are on the axis
Answer & Solution
Correct Answer
(D) a family of circles whose centres are on the axis
Step-by-step Solution
Detailed explanation
Given differential equation is
\[
\begin{aligned}
y \frac{d y}{d x}+x &=c \\
\Rightarrow \quad y d y &=(c-x) d x
\end{aligned}
\]
On integrating both sides, we get
\[
\begin{aligned}
\frac{y^{2}}{2} &=c x-\frac{x^{2}}{2}+d \\
\Rightarrow \quad y^{2}+x^{2}-2 c x-2 d &=0
\end{aligned}
\]
Hence, it represents a family of circles whose centres are on the \(x\)-axis.
\[
\begin{aligned}
y \frac{d y}{d x}+x &=c \\
\Rightarrow \quad y d y &=(c-x) d x
\end{aligned}
\]
On integrating both sides, we get
\[
\begin{aligned}
\frac{y^{2}}{2} &=c x-\frac{x^{2}}{2}+d \\
\Rightarrow \quad y^{2}+x^{2}-2 c x-2 d &=0
\end{aligned}
\]
Hence, it represents a family of circles whose centres are on the \(x\)-axis.
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