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(B) Let the equation of the straight line be $y = kx + c$,where $k$ and $c$ are constants. Then,$\frac{dy}{dx} = k$. Substituting this into the given

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$2$

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(B) Let the equation of the straight line be $y = kx + c$,where $k$ and $c$ are constants. Then,$\frac{dy}{dx} = k$. Substituting this into the given differential equation $\frac{dy}{dx} + x \left( \frac{dy}{dx} \right)^2 - y = 0$,we get: $k + xk^2 - (kx + c) = 0$ $k + xk^2 - kx - c = 0$ $(k^2 - k)x + (k - c) = 0$ For this to hold for all $x$,the coefficients must be zero: $k^2 - k = 0 \Rightarrow k(k - 1) = 0 \Rightarrow k = 0$ or $k = 1$. If $k = 0$,then $k - c = 0 \Rightarrow c = 0$. The line is $y = 0$. If $k = 1$,then $k - c = 0 \Rightarrow c = 1$. The line is $y = x + 1$. Thus,there are $2$ such straight lines.

Number of straight lines which satisfy the differential equation $\frac{dy}{dx} + x \left( \frac{dy}{dx} \right)^2 - y = 0$ is:

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