JEE Mains · Maths · STD 12 - 9. differential equations
If the general solution of the differential equation \(y' = \frac{y}{x} + \phi \left( {\frac{x}{y}} \right)\) , for some function \(\phi \), is given by \(y \ln \,\left| {cx} \right| = x\), where \(c\) is an arbitrary constant, then \(\phi \,(2)\) is equal to:
- A \(4\)
- B \(\frac{1}{4}\)
- C \(-4\)
- D \(-\frac{1}{4}\)
Answer & Solution
Correct Answer
(D) \(-\frac{1}{4}\)
Step-by-step Solution
Detailed explanation
\(\text { Given } \frac{d y}{d x}=\frac{y}{x}+\phi\left(\frac{y}{x}\right)\) ....\((1)\) Let \(\left(\frac{y}{x}\right)=v\) so that \(y=x v\) or \(\frac{d y}{d x}=x \frac{d h}{d x}+y\) ....\((2)\) from \((1)\) and \((2), x \frac{d v}{d x}+v=v+\phi\left(\frac{1}{v}\right)\) or.…
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