GATE 2021 CE (CE2) – Question 36

Engineering Mathematics · ODE: First order and higher order linear equations, Euler-Cauchy equations, initial and boundary value problems · 2 marks · Multiple choice

For constant k, the general solution of $dy/dx-y/x=1$ has which form?

$y=x\ln(kx)$
$y=k\ln(x)$
$y=x\ln(x)$
$y=xk\ln(k)$

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Correct answer: (A) $y=x\ln(kx)$

Explanation

Divide by x to obtain $d(y/x)/dx=1/x$. Integration gives $y/x=\ln|x|+C$, or $y=x\ln|x|+Cx$. On a positive-domain interval this can be written **x ln(kx), A**. Option C is only one particular choice of the arbitrary constant.