KCET · Maths · Differential Equations
The differential equation of the family of straight lines whose slope is equal to \(y\)-intercept is
- A \((x+1) \frac{d y}{d x}-y=0\)
- B \((x+1) \frac{d y}{d x}+y=0\)
- C \(\frac{d y}{d x}=\frac{x-1}{y-1}\)
- D \(\frac{d y}{d x}=\frac{x+1}{y+1}\)
Answer & Solution
Correct Answer
(A) \((x+1) \frac{d y}{d x}-y=0\)
Step-by-step Solution
Detailed explanation
Equation of line whose slope is equal to \(y\) intercept, is
\[
\Rightarrow \quad \begin{aligned}
y &=c x+c=c(x+1) \\
\frac{d y}{d x} &=c
\end{aligned}
\]
On putting the value of \(c\) from Eq. (i) in Eq. (ii), we get \(\frac{d y}{d x}=\frac{y}{x+1}\)
\[
\Rightarrow \quad(x+1) \frac{d y}{d x}-y=0
\]
which is the required differential equation
\[
\Rightarrow \quad \begin{aligned}
y &=c x+c=c(x+1) \\
\frac{d y}{d x} &=c
\end{aligned}
\]
On putting the value of \(c\) from Eq. (i) in Eq. (ii), we get \(\frac{d y}{d x}=\frac{y}{x+1}\)
\[
\Rightarrow \quad(x+1) \frac{d y}{d x}-y=0
\]
which is the required differential equation
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