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GATE 2020 EE – Question 26

Engineering Mathematics · Differential Equations: First order equations, Higher order linear differential equations, Variation of parameters, Cauchy and Euler equations, Partial differential equations, Separation of variables · 1 mark · Numerical answer

Consider the initial value problem below. The value of $y$ at $x=\ln2$, (rounded off to 3 decimal places) is ________.

$$\frac{dy}{dx}=2x-y,\qquad y(0)=1$$

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Correct answer: 0.8774 to 0.8952

Explanation

The equation $\frac{dy}{dx}+y=2x$ has integrating factor $e^x$, so $ye^x=\int2xe^x\,dx=2(x-1)e^x+C$. With $y(0)=1$, $C=3$, so $y=2x-2+3e^{-x}$. At $x=\ln2$: $y=2(0.6931)-2+1.5=0.886$.