GATE 2019 CE (CE1) – Question 54

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

For $x^2y^{\prime\prime}-2xy^\prime+2y=0$, with y(1) = 0 and y(2) = 2, find y(3) to one decimal place.

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Correct answer: 5.9 to 6.1

Explanation

Use the Euler trial $y=x^m$, giving $m(m-1)-2m+2=(m-1)(m-2)=0$. Thus $y=C_1x+C_2x^2$. Conditions imply $C_1+C_2=0$ and $2C_1+4C_2=2$, hence $C_1=-1$, $C_2=1$. Then $y(3)=9-3=$ **6.0**.