Chapter 13: Problem 1109
The integrating factor of the differential equation \((d y / d x) \cdot(x \log x)+y=2 \log x\) is (A) \(\mathrm{e}^{\mathrm{x}}\) (B) \(\log x\) (C) \(\log (\log \mathrm{x})\) (D) \(\mathrm{x}\)
Chapter 13: Problem 1109
The integrating factor of the differential equation \((d y / d x) \cdot(x \log x)+y=2 \log x\) is (A) \(\mathrm{e}^{\mathrm{x}}\) (B) \(\log x\) (C) \(\log (\log \mathrm{x})\) (D) \(\mathrm{x}\)
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Get started for freeFamily of curves \(\mathrm{y}=\mathrm{e}^{\mathrm{x}}(\mathrm{A} \cos \mathrm{x}+\mathrm{B} \sin \mathrm{x})\) represents the differential equation: \(\quad\) (where \(\mathrm{A}\) and \(\mathrm{B}\) are arbitrary constant) (A) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)-2(\mathrm{dy} / \mathrm{dx})+\mathrm{y}=0\) (B) \(\left(d^{2} y / d x^{2}\right)-2(d y / d x)-2 y=0\) (C) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)-2(\mathrm{dy} / \mathrm{dx})-\mathrm{y}=0\) (D) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right)-2(\mathrm{dy} / \mathrm{dx})+2 \mathrm{y}=0\)
Solution of \(\left(d^{2} y / d x^{2}\right)=\log x\) is: (A) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}-(3 / 4) \mathrm{x}^{2}+\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (B) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}+(3 / 4) \mathrm{x}^{2}+\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (C) \(\mathrm{y}=(1 / 2) \mathrm{x}^{2} \log \mathrm{x}-(3 / 4) \mathrm{x}^{2}-\mathrm{c}_{1} \mathrm{x}+\mathrm{c}_{2}\) (D) None of these
If the general solution of \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / \mathrm{x})+\mathrm{f}(\mathrm{x} / \mathrm{y})\) is \(\mathrm{y}\) \(=[\mathrm{x} / \log |\mathrm{cx}|]\), then \(\mathrm{f}(\mathrm{x} / \mathrm{y})\) is given by: (A) \(\left(\mathrm{x}^{2} / \mathrm{y}^{2}\right)\) (B) \(\left(\mathrm{y}^{2} / \mathrm{x}^{2}\right)\) (C) \(\left[\left(-\mathrm{x}^{2}\right) / \mathrm{y}^{2}\right]\) (D) \(\left[\left(-\mathrm{y}^{2}\right) / \mathrm{x}^{2}\right]\)
If \(\sin (x+y)(d y / d x)=5\) then (A) \(5 \int[(\mathrm{dt}) /(5+\sin \mathrm{t})]=\mathrm{t}+\mathrm{x}(\) where \(\mathrm{t}=\mathrm{x}+\mathrm{y})\) (B) \(5 \int[(\mathrm{dt}) /(5+\sin \mathrm{t})]=\mathrm{t}-\mathrm{x}(\) where \(\mathrm{t}=\mathrm{x}+\mathrm{y})\) (C) \(\int[(\mathrm{dt}) /(5+\operatorname{cosec} \mathrm{t})]=\mathrm{d} \mathrm{x}(\) where \(\mathrm{t}=\mathrm{x}+\mathrm{y})\) (D) \(\int[(\mathrm{dt}) /(5 \sin t+1)]=\mathrm{dt}(\) where \(t=x+y)\)
If \((\mathrm{dy} / \mathrm{dx})=\mathrm{y}+3>0\) and \(\mathrm{y}(0)=2\), then \(\mathrm{y}(\log 2)\) is equal to. \(\begin{array}{ll}\text { (A) }-2 & \text { (B) } 5\end{array}\) (C) 7 (D) 13
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