JEE Advanced · 2006 · 09 Apr 2006 · Differential Equations

More JEE Advanced questions on Differential Equations

1. Let $\mathrm{y}:(-\infty ,\infty )\to (0,\infty )$ be the solution of the differential equation
   

   $$\frac{dy}{dx}=\frac{e^{5x}{y}^3+y^3}{e^x+e^x{y}^4}$$

   
   satisfying $\mathrm{y}(0)=\frac{1}{\sqrt{2}}$. Then the value of $\mathrm{y}\left({\log }_{\mathrm{e}}2\right)$ …
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2. Let $\mathrm{y}=f(\mathrm{x})$ be the real valued function defined on the interval $(0,\infty )$, satisfying $\mathrm{y}(1)=0$ and the differential equation
   

   $$x\frac{dy}{dx}=y-x^3$$

   
   Then which of the following statements is (are) TRUE ?
   (A) The function $f$ has a local minim …
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3. For all $x>0$, let $y_1(x),y_2(x)$, and $y_3(x)$ be the functions satisfying …
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4. Let $y(x)$ be the solution of the differential equation
   

   $$x^2\frac{dy}{dx}+xy=x^2+y^2,x>\frac{1}{e},$$

   
   satisfying $y(1)=0$. Then the value of $2\frac{(y(e){)}^2}{y\left(e^2\right)}$ is ____.
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5. Let $f(\mathrm{x})$ be a continuously differentiable function on the interval $(0,\infty )$ such that $f(1)=2$ and ${\lim }_{\mathrm{t}\to \mathrm{x}}\frac{{\mathrm{t}}^{10}f(\mathrm{x})-{\mathrm{x}}^{10}f(\mathrm{t})}{{\mathrm{t}}^9-{\mathrm{x}}^9}=1$ for each …
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6. Let $\mathrm{f}:[1,\infty )\to \mathbb{R}$ be a differentiable function such that $\mathrm{f}(1)=\frac{1}{3}$ and $3{\int }_1^{\mathrm{x}}\mathrm{f}(\mathrm{t})\mathrm{dt}=\mathrm{xf}(\mathrm{x})-\frac{{\mathrm{x}}^3}{3},\mathrm{x}\in [1,\infty )$. Let $e$ denote the …
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