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Differential Equations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations1 Mark
MCQ
The solution of the differential equation $2\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=3$ resresents:
  • A
    circles
  • B
    straight lines
  • C
    ellipses
  • ✓
    parabolas
✓

Answer

Correct option: D.
parabolas
We have,
$2\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=3$
$\Rightarrow 2\text{x}\frac{\text{dy}}{\text{dx}}=3+\text{y}$
$\Rightarrow \frac{1}{3+\text{y}}\text{dy}=\frac{1}{2\text{x}}\text{dx}$
Interating both sides, we get
$\Rightarrow \int\frac{1}{3+\text{y}}\text{dy}=\frac{1}{2}\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow \log|3+\text{y}|=\frac{1}{2}\log|\text{x}|+\log|\text{C}|$
$\Rightarrow\log|\frac{3+\text{y}}{\sqrt{\text{x}}}|=\log\text{C}$
$\Rightarrow\frac{3+\text{y}}{\sqrt{\text{x}}}=\text{C}$
$\Rightarrow 3+\text{y}=\text{C}\sqrt{\text{x}}$
Squaring both sides, we get
$(3+\text{y})^{2}=\text{C}{\text{x}}\ ...(\text{i})$
Thus, $(i)$ the equation of parabolas.

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