Question

\[2x\frac{dy}{dx} = 5y, y\left( 1 \right) = 1\]

Answer 1

\[2x\frac{dy}{dx} = 5y, y\left( 1 \right) = 1\]
\[ \Rightarrow \frac{2}{y}dy = \frac{5}{x} dx\]
Integrating both sides, we get 
\[2\int\frac{1}{y}dy = 5\int\frac{1}{x} dx\]
\[ \Rightarrow 2\log \left| y \right| = 5\log \left| x \right| + C . . . . . (1)\]
We know that at x = 1 and y = 1 . 
Substituting the values of x and y in (1), we get
\[2\log \left| 1 \right| = 5\log \left| 1 \right| + C\]
\[ \Rightarrow C = 0\]
Substituting the value of C in (1), we get
\[2 \log \left| y \right| = 5 \log \left| x \right| + 0\]
\[ \Rightarrow y = \left| x \right|^\frac{5}{2} \]
\[\text{ Hence, }y = \left| x \right|^\frac{5}{2}\text{ is the required solution .}\]