KCET · Maths · Application of Derivatives
The integrating factor of the differential equation \( x \cdot \frac{d y}{d x}+2 y=x^{2} \) is \( (x \neq 0) \)
- A \( x^{2} \)
- B log \( \mid x \)
- C \( e^{\log x} \)
- D \( x \)
Answer & Solution
Correct Answer
(A) \( x^{2} \)
Step-by-step Solution
Detailed explanation
Given differential equation
\[
\begin{array}{l}
x \frac{d y}{d x}+2 y=x^{2} \\
\Rightarrow \frac{d y}{d x}+\frac{2}{x} y=x
\end{array}
\]
The general differential equation of this type is given by
\[
\frac{d y}{d x}+P(x) y=Q(x)
\]
So, I.F. factor is given by \( I \cdot F \cdot=e^{\int P d x} \)
\[
\begin{array}{l}
=e^{\int \frac{2}{x} d x}=e^{2 \log x} \\
=e^{\log x^{2}}=x^{2}
\end{array}
\]
\[
\begin{array}{l}
x \frac{d y}{d x}+2 y=x^{2} \\
\Rightarrow \frac{d y}{d x}+\frac{2}{x} y=x
\end{array}
\]
The general differential equation of this type is given by
\[
\frac{d y}{d x}+P(x) y=Q(x)
\]
So, I.F. factor is given by \( I \cdot F \cdot=e^{\int P d x} \)
\[
\begin{array}{l}
=e^{\int \frac{2}{x} d x}=e^{2 \log x} \\
=e^{\log x^{2}}=x^{2}
\end{array}
\]
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