Question

Number of straight lines which satisfy the differential equation $\dfrac{{dy}}{{dx}} + x{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - y = 0$ is:
(A) $1$
(B) $2$
(C) $3$
(D) $4$

Answer 1

In this question we will make use of the general equation of straight line and then we will differentiate the general equation of straight line to find the slope and then we will substitute the value of the slope in the differential equation given in the question and then we will find the final answer.

Formula used:
We will use the general equation of straight line $y = mx + c$ where $m$ is the slope and $c$ is the $y$ intercept.

Complete step-by-step answer:
The differential equation given to us is \dfrac{{dy}}{{dx}} + x{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - y = 0\,\\_\\_\\_\left( 1 \right)
We know that the general equation of straight line is given by y = mx + c\\_\\_\\_\left( 2 \right)
Now, differentiate the equation $\left( 2 \right)$ with respect to $x$ . Therefore the equation $\left( 2 \right)$ can be written as $\dfrac{{dy}}{{dx}} = m$
Hence, we got the value of the slope and now substitute the value of the slope in equation $\left( 1 \right)$ . Therefore the differential equation can be written as:
$\Rightarrow m + x{m^2} - mx - c = 0$
Take $x$ common from the above equation
$\Rightarrow x\left( {{m^2} - m} \right) + m - c = 0$
Now, from the above equation we can say that if $x\left( {{m^2} - m} \right) = 0$ and $m - c = 0$ then only $x\left( {{m^2} - m} \right) + m - c = 0$ . Therefore, we can write $x\left( {{m^2} - m} \right) = 0$ . Now, from the equation $x\left( {{m^2} - m} \right) = 0$ we can find the value of $m$ . The value of $m$ is $0$ and $1$ .
Now, if $m = 0$ then $c = 0$ . Similarly, if $m = 1$ then $c = 1$ .
Hence, one of the equation of straight line is $y = 0$ and the other equation of line is $y = x + 1$ . Therefore, there are two straight lines which satisfies the differential equation $\dfrac{{dy}}{{dx}} + x{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - y = 0$ .
Hence, the correct option is (B) .

Note: In this question the important thing is that we should be able to find the value of the slope from the general equation of the straight line and the other important thing is to find the equation of lines with the help of the value of the slope and the differential equation.