MHT CET · Maths · Differential Equations
Integrating factor of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+y=\frac{1+y}{x}\) is
- A \(\frac{x}{\mathrm{e}^x}\)
- B \(x e^x\)
- C \(\mathrm{e}^x\)
- D \(\frac{\mathrm{e}^x}{x}\)
Answer & Solution
Correct Answer
(D) \(\frac{\mathrm{e}^x}{x}\)
Step-by-step Solution
Detailed explanation
\(\begin{array}{ll} & \frac{\mathrm{d} y}{\mathrm{~d} x}+y=\frac{1+y}{x} \\ & \Rightarrow \frac{\mathrm{~d} y}{\mathrm{~d} x}+y=\frac{1}{x}+\frac{y}{x} \\ & \Rightarrow \frac{\mathrm{~d} y}{\mathrm{~d} x}+\left(1-\frac{1}{x}\right) y=\frac{1}{x} \\ \therefore \quad & \text { I.F. }=\mathrm{e}^{\int\left(1-\frac{1}{x}\right) \mathrm{dx}}=\mathrm{e}^{x-\log x}=\frac{\mathrm{e}^x}{x}\end{array}\)
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