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Differential Equations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations1 Mark
MCQ
The equation of the curve whose slope is given by $\frac{\text{dy}}{\text{dx}}=\frac{2\text{y}}{\text{x}};\text{x}>0,\text{y}>0$ and which passes through the point $(1, 1)$ is:
  • ✓
    $\text{x}^{2}=\text{y}$
  • B
    $\text{y}^{2}=\text{x}$
  • C
    $\text{x}^{2}=2\text{y}$
  • D
    $\text{y}^{2}=2\text{x}$
✓

Answer

Correct option: A.
$\text{x}^{2}=\text{y}$
We have,
$\frac{\text{dy}}{\text{dx}}=\frac{2\text{y}}{\text{x}}$
$\Rightarrow\frac{1}{2}\times\frac{1}{\text{y}}\text{dy}=\frac{1}{\text{x}}\text{dx}$
Interating both sides, we get
$\Rightarrow\frac{1}{2}\int\frac{1}{\text{y}}\text{dy}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow\frac{1}{2}\ \log{\text{y}}=\log{\text{x}}+\log\text{C}$
$\Rightarrow\log{\text{y}}^{\frac{1}{2}}-\log{\text{x}}=\log\text{C}$
$\Rightarrow\log\big(\frac{\sqrt{\text{y}}}{2}\big)=\log\text{C}$
$\Rightarrow\frac{\sqrt{\text{y}}}{2}=\text{C}$
$\Rightarrow\sqrt{\text{y}}=\text{Cx}\ ...(\text{i})$
As $(i)$ passes through $(1, 1)$, we get
$1=\text{C}$
Putting the value of $C$ in $(1)$, we get
$\Rightarrow\sqrt{\text{y}}=\text{x}$
$\Rightarrow{\text{y}}=\text{x}^{2}$

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