Which of the following is a second order differential equation?

MCQ

A
$\left(y^{\prime}\right)^2+x=y^2$
✓
$y^{\prime} y^{\prime \prime}+y=\sin x$
C
$y^{\prime \prime \prime}+\left(y^{\prime \prime}\right)^2+y=0$
D
$y^{\prime}=y^2$

✓

Answer

Correct option: B.

$y^{\prime} y^{\prime \prime}+y=\sin x$

(b) : (a) $\left(\frac{d y}{d x}\right)^2+x=y^2$; order $=1$
(b) $\left(\frac{d y}{d x}\right)\left(\frac{d^2 y}{d x^2}\right)+y=\sin x ;$ order $=2$
(c) $\frac{d^3 y}{d x^3}+\left(\frac{d^2 y}{d x^2}\right)^2+y=0 ;$ order $=3$
(d) $\frac{d y}{d x}=y^2 ;$ order $=1$

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Let $f _1: R \rightarrow R , f _2:[0, \infty) \rightarrow R , f _3: R \rightarrow R$ and $f _4: R \rightarrow[0, \infty)$ be defined by

$f_1(x)=\left\{\begin{array}{lll}|x| & \text { if } & x<0, \\ e^x & \text { if } & x \geq 0 ;\end{array}\right.$

$f_2(x)=x^2$

$f_3(x)=\left\{\begin{array}{ccc}\sin x & \text { if } & x < 0, \\ x & \text { if } & x \geq 0\end{array}\right.$ and

$f_4(x)=\left\{\begin{array}{ccc}f_2\left(f_1(x)\right) & \text { if } & x < 0, \\ f_2\left(f_1(x)\right)-1 & \text { if } & x \geq 0\end{array}\right.$

List $I$	List $II$
$P.$ $ f_4$ is	$1.$ onto but not one-one
$Q.$ $f_3$ is	$2.$ neither continuous nor one-one
$R.$ $f _2 \circ f _1$ is	$3.$ differentiable but not one-one
$S.$ $ f_2$ is	$4.$ continuous and one-one