The general solution of the differential equation : \( x\displaystyle\frac{dy}{dx}-ay=1\ (a\ne 0) \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( y=-\displaystyle\frac{1}{a}+Cx^{a} \) : Where C is an arbitrary constant
Dividing by x gives the linear form \( \frac{dy}{dx}+Py=Q \) (taking x > 0):
\( \displaystyle \frac{dy}{dx}-\frac{a}{x}\,y=\frac{1}{x} \)
\( \displaystyle \text{I.F.}=e^{\int -\frac{a}{x}\,dx} \)
\( \displaystyle =e^{-a\log x}=x^{-a} \)
Multiplying by the integrating factor and integrating:
\( \displaystyle y\cdot x^{-a}=\int\frac{1}{x}\cdot x^{-a}\,dx \)
\( \displaystyle =\int x^{-a-1}\,dx \)
\( \displaystyle =\frac{x^{-a}}{-a}+C \)
Multiplying through by \( x^{a} \):
\( \displaystyle y=-\frac{1}{a}+Cx^{a} \)
Shortcut: Option (1) can be checked directly: \( y=-\frac{1}{a}+Cx^{a} \) gives \( x\frac{dy}{dx}=aCx^{a} \), so
\( \displaystyle x\frac{dy}{dx}-ay=aCx^{a}+1-aCx^{a} \)
\( \displaystyle =1 \)