Solve the following differential equations: 2(y+3)-xy dy dx =0,y(… - Vidyadip

Question

Solve the following differential equations:
$2(\text{y}+3)-\text{xy}\frac{\text{dy}}{\text{dx}}=0,\text{y}(1)=-2$

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Answer

$2(\text{y}+3)-\text{xy}\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow2(\text{y}+3)=\text{xy}\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\frac{2}{\text{x}}\text{dx}=\frac{\text{y}}{\text{y}+3}\text{dy}$
$\Rightarrow\frac{2}{\text{x}}\text{dx}=\frac{\text{y}+3-3}{\text{y}+3}\text{dy}$
$\Rightarrow\frac{2}{\text{x}}\text{dx}=\Big(1-\frac{3}{\text{y}+3}\Big)\text{dy}$
$\Rightarrow\int\frac{2}{\text{x}}\text{dx}=\int\Big(1-\frac{3}{\text{y}+3}\Big)\text{dy}$
$\Rightarrow2\log\text{x = y}-3\log|\text{y}+3|+\text{C}$
$\Rightarrow\log\text{x}^2+\log|(\text{y}+3)^3|=\text{y + C}$
$\Rightarrow\log|(\text{x}^2)(\text{y}+3)^3|=\text{y + C}...(1)$
$\Rightarrow\log|(1)^2(-2+3)^3|=-2+\text{C}$
$\Rightarrow\text{C}=2$
Substituting the value of C in (1), we get
$\log|(\text{x}^2)(\text{y}+3)^3|=\text{y}+2$
$\Rightarrow(\text{x}^2)(\text{y}+3)^3=\text{e}^{\text{y}+2}$

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