KCET · Maths · Differential Equations
The general solution of the differential equation \(2 x \frac{d y}{d x}-y=3\) is a family of
- A hyperbolas
- B parabolas
- C straight lines
- D circles
Answer & Solution
Correct Answer
(B) parabolas
Step-by-step Solution
Detailed explanation
The given differential equation is
\(2 x \frac{d y}{d x}-y=3\)
\(\Rightarrow \quad 2 x \frac{d y}{d x}=(y+3)\)
\(\Rightarrow \quad 2 \int \frac{\mathrm{dy}}{(\mathrm{y}+3)}=\int \frac{\mathrm{dx}}{\mathrm{x}}\) (on integrating)
\(\Rightarrow \quad 2 \log (y+3)=\log x+\log c\)
\(\Rightarrow \quad(\mathrm{y}+3)^{2}=\mathrm{cx}\)
which represents a family of parabolas.
\(2 x \frac{d y}{d x}-y=3\)
\(\Rightarrow \quad 2 x \frac{d y}{d x}=(y+3)\)
\(\Rightarrow \quad 2 \int \frac{\mathrm{dy}}{(\mathrm{y}+3)}=\int \frac{\mathrm{dx}}{\mathrm{x}}\) (on integrating)
\(\Rightarrow \quad 2 \log (y+3)=\log x+\log c\)
\(\Rightarrow \quad(\mathrm{y}+3)^{2}=\mathrm{cx}\)
which represents a family of parabolas.
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