GATE 2019 EC – Question 53

Engineering Mathematics · Differential Equations · 2 marks · Numerical answer

Consider the homogeneous ordinary differential equation

$$x^2\frac{d^2y}{dx^2}-3x\frac{dy}{dx}+3y=0,\qquad x>0$$

with $y(x)$ as a general solution. Given that $y(1)=1$ and $y(2)=14$, the value of $y(1.5)$, rounded off to two decimal places, is ________.

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Correct answer: 5.22 to 5.28

Explanation

This is a Cauchy-Euler equation; trying $y=x^m$ gives $m^2-4m+3=0$, so $m=1,3$ and $y=ax+bx^3$. The conditions $a+b=1$ and $2a+8b=14$ give $b=2$, $a=-1$. So $y(1.5)=-1.5+2(3.375)=5.25$.