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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations1 Mark
MCQ
The differential equation satisfied by $\text{ax}^{2}+\text{by}^{2}=1$ is :
  • A
    $\text{xyy}_{2}+\text{y}_{1}^{2}+\text{yy}_{1}=0$
  • ✓
    $\text{xyy}_{2}+\text{xy}_{1}^{2}-\text{yy}_{1}=0$
  • C
    $\text{xyy}_{2}+\text{xy}_{1}^{2}+\text{yy}_{1}=0$
  • D
    None of these.
✓

Answer

Correct option: B.
$\text{xyy}_{2}+\text{xy}_{1}^{2}-\text{yy}_{1}=0$
We have,
$\text{ax}^{2}+\text{by}^{2}=1\ ...(\text{i})$
Differential both sides of $(i)$ with $x,$ we get
$2\text{ax}+2\text{by}\frac{\text{dy}}{\text{dy}}=0\ ...(\text{ii})$
Differential both sides of $(ii)$ with $x,$ we get
$2\text{ax}+2\text{b}\Big(\frac{\text{dy}}{\text{dy}}\Big)^{2}+2\text{}by\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=0$
$\Big[\text{y}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}\Big(\frac{\text{dy}}{\text{dx}^{2}}\Big)\Big]=-\frac{2\text{a}}{2\text{b}}$
$\text{x}\Big[\text{y}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}\Big(\frac{\text{dy}}{\text{dx}^{2}}\Big)\Big]=-\Big(-\frac{\text{y}}{\text{x}}\frac{\text{dy}}{\text{dx}}\Big)$
$\text{xy}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{x}\Big(\frac{\text{dy}}{\text{dx}}\big)^{2}-\text{y}\frac{\text{dy}}{\text{dx}}=0$
$\text{xyy}_{2}+\text{x}(\text{y}_{1}^{2})-\text{yy}_{1}=0$

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