question_answer

Show that the differential equation that represents the family of all parabolas having their axis of symmetry coincident with the axis of x is \[y{{y}_{2}}+y_{1}^{2}=0.\]

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.