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Differential Equations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations2 Marks
MCQ
The differential equation of all parabolas whose axis is $y$-axis is
  • A
    $x \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$
  • B
    $x \frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$
  • C
    $\frac{d^2 y}{d x^2}-y=0$
  • D
    $\frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$
✓

Answer

(a) : Consider a parabola having its vertex at $(0, k)$ and axis along $y$-axis.
Equation of parabola is
$
\begin{aligned}
& (x-0)^2=4 a(y-k) \\
\Rightarrow \quad & x^2=4 a y-4 a k
\end{aligned}
$
Differentiating w.r.t $x$, we get
$
2 x=4 a \frac{d y}{d x} \Rightarrow x=2 a \frac{d y}{d x} \Rightarrow \frac{1}{2 a}=\frac{1}{x} \frac{d y}{d x}
$
Again differentiating w.r.t. $x$, we get,
$
\frac{1}{x} \cdot \frac{d^2 y}{d x^2}+\frac{d y}{d x}\left(-\frac{1}{x^2}\right)=0
$
$\Rightarrow \quad x \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$ is the required differential equation.


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