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Question
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
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Solution
We have,
\[y = Cx + 5 . . . . . \left( 1 \right)\]
\[ \Rightarrow \frac{dy}{dx} = C\]
\[\text{ Substituting the value of C in }\left( 1 \right),\text{ we get }\]
\[y = \frac{dy}{dx} \times x + 5\]
\[ \Rightarrow x\frac{dy}{dx} - y + 5 = 0 \]
\[\text{ Hence, }x\frac{dy}{dx} - y + 5 = 0\text{ is the differential equation representing the family of straight lines }y = Cx + 5, \text{ where C is an arbitary constant . }\]
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