Answer:
The equation that represents a family of parabolas having their axis of symmetry coincident with the axis of x is \[{{y}^{2}}=4a(x-h)\] ...(i) Where, a and h are parameters. This equation contains two parameters a and h, so we will differentiate it twice to obtain a second differential eqution. On differentiating Eq. (i) w.r.t. x, we get \[2y\frac{dy}{dx}=4a\] \[\Rightarrow \] \[y\frac{dy}{dx}=2a\] ?(ii) On differentiating Eq. (ii) w.r.t. to x, we get \[y\frac{{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}}=0\] \[\Rightarrow \] \[y{{y}_{2}}+y_{1}^{2}=0\] Which is the required differential equation.
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