MCQ
The differential equation whose solution is $A{x^2} + B{y^2} = 1$ where $A$ and $B$ are arbitrary constants is of
- Asecond order and second degree
- Bfirst order and second degree
- Cfirst order and first degree
- ✓second order and first degree
$A x+b y \frac{d y}{d x}=0 \quad \ldots(2)$
$A+B y \frac{d^{2} y}{d x^{2}}+B\left(\frac{d y}{d x}\right)^{2}=0 \quad \ldots(3)$
From $(2)$ and $(3)$
$x\left\{-B y \frac{d^{2} y}{d x^{2}}-B\left(\frac{d y}{d x}\right)^{2}\right\}+B y \frac{d y}{d x}=0$
Dividing both sides by $-B,$ we get
$\Rightarrow x y \frac{d^{2} y}{d x^{2}}+x\left(\frac{d y}{d x}\right)^{2}-y \frac{d y}{d x}=0$
Which is $D E$ of order $2$ and degree $1$
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$f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x<1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x<2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.$
where a,b,c $\in R$ and $[t]$ denotes greatest integer less than or equal to $t.$ Then, which of the following statements is true $?$
$f\left( x \right) = \int_1^x {\left\{ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right\}} dt$ is maximum when $x$ is equal to