Fill in the blanks. The solution of differential equation dx=x dy… - Vidyadip

Question

Fill in the blanks.
The solution of differential equation $\cot\text{y dx}=\text{x dy} $ is _________.

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Answer

The solution of differential equation $\cot\text{y dx}=\text{x dy} $ is $\text{x}=\text{C}\sec\text{ y}.$Solution:
Given differential equation is $\cot\text{y dx}=\text{x dy} $ $\Rightarrow\frac{1}{\text{x}}\text{dx}=\tan\text{y dy}$ On integrating both sides, we get $\Rightarrow\int\frac{1}{\text{x}}\text{dx}=\int\tan\text{y dy}$ $\Rightarrow\log(\text{x})=\log(\sec\text{y})+\log\text{C}$ $\Rightarrow\log\Big(\frac{\text{x}}{\sec\text{y}}\Big)=\log\text{C}$ $\Rightarrow\frac{\text{x}}{\sec\text{y}}=\text{C}$ $\Rightarrow\text{x}=\text{C}\sec\text{y}$

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