Question:
What is the general solution of the differential equation \(y\,dx – (x + 2y^2)dy = 0\)?
अवकल समीकरण \(y\,dx – (x + 2y^2)dy = 0\) का सामान्य हल क्या है?
Options:
- \(x = y^2 + cy\)
- \(x = 2cy^2\)
- \(x = 2y^2 + cy\)
- None of the above
- \(x = y^2 + cy\)
- \(x = 2cy^2\)
- \(x = 2y^2 + cy\)
- उपरोक्त में से कोई नहीं
Correct Answer: c)
Explanation:
Rearrange the differential equation:
\[y\,dx – x\,dy = 2y^2\,dy\]
Dividing both sides by \(y^2\):
\[\frac{y\,dx – x\,dy}{y^2} = 2\,dy\]
Recognizing the left-hand side as the derivative of a quotient:
\[d\left(\frac{x}{y}\right) = 2\,dy\]
Integrating both sides:
\[\frac{x}{y} = 2y + c \implies x = 2y^2 + cy\]
Hence, the correct option is (c).
अवकल समीकरण को पुनर्व्यवस्थित करने पर:
\[y\,dx – x\,dy = 2y^2\,dy\]
दोनों पक्षों को \(y^2\) से विभाजित करने पर:
\[\frac{y\,dx – x\,dy}{y^2} = 2\,dy \implies d\left(\frac{x}{y}\right) = 2\,dy\]
दोनों पक्षों का समाकलन करने पर:
\[\frac{x}{y} = 2y + c \implies x = 2y^2 + cy\]
अतः सही विकल्प (c) है।