Question

The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution

Options

• y − x³ = 2cx

• 2y − x³ = cx

• 2y + x² = 2cx

• y + x² = 2cx

Answer 1

2y − x³ = cx

 

We have,
\[x\frac{dy}{dx} - y = x^2\]
\[\Rightarrow \frac{dy}{dx} - \frac{1}{x}y = x^2 \]
\[\text{ Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]
\[P = - \frac{1}{x} \]
\[Q = x^2 \]
Now, 
\[I . F . = e^{- \int\frac{1}{x}dx} = e^{- \log\left| x \right|} \]
\[ = e^{log\left| \frac{1}{x} \right|} \]
\[ = \frac{1}{x}\]
\[y \times I . F = \int x^2 \times I . Fdx + C\]
\[ \Rightarrow y\frac{1}{x} = \int x^2 \times \frac{1}{x}dx + C\]
\[ \Rightarrow y\frac{1}{x} = \int xdx + C\]
\[ \Rightarrow y\frac{1}{x} = \frac{x^2}{2} + C\]
\[ \Rightarrow 2y - x^3 = Cx\]