KCET · Maths · Differential Equations
The solution of the differential equation \( x \frac{d y}{d x}-y=3 \) represents a family of
- A Straight lines
- B Circles
- C Parabolas
- D Ellipses
Answer & Solution
Correct Answer
(A) Straight lines
Step-by-step Solution
Detailed explanation
Given differential equation is,
\[
\begin{array}{l}
x \frac{d y}{d x}-y=3 \\
\Rightarrow x \frac{d y}{d x}=3+y \\
\Rightarrow \frac{d y}{3+y}=\frac{d x}{x}
\end{array}
\]
Integrating both the sides, we have
\[
\begin{array}{l}
\int \frac{d y}{3+y}=\int \frac{d x}{x} \\
\Rightarrow \ln (3+y)=\ln x+\ln c \\
\Rightarrow(3+y)=c x
\end{array}
\]
The equation is a straight line equation.
\[
\begin{array}{l}
x \frac{d y}{d x}-y=3 \\
\Rightarrow x \frac{d y}{d x}=3+y \\
\Rightarrow \frac{d y}{3+y}=\frac{d x}{x}
\end{array}
\]
Integrating both the sides, we have
\[
\begin{array}{l}
\int \frac{d y}{3+y}=\int \frac{d x}{x} \\
\Rightarrow \ln (3+y)=\ln x+\ln c \\
\Rightarrow(3+y)=c x
\end{array}
\]
The equation is a straight line equation.
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