The differential equation which represents the family of curves y = c_1 e^c_2 x , where c_1 and c_2 are arbitrary constants is: - Mathematics MCQ AIEEE 2009 2026 | ExamTest.live

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The differential equation which represents the family of curves y = c_{1} e^{c_{2} x}, where c_{1} and c_{2} are arbitrary constants is:

Solution & Step-by-step Explanation

y = c_{1} e^{c_{2} x} .Differentiating w.r.t x : y^{'} = c_{1} c_{2} e^{c_{2} x} = c_{2} y .Differentiating again: y^{''} = c_{2} y^{'} .From the first derivative, c_{2} = y^{'} / y .Substitute c_{2} in second derivative: y^{''} = (y^{'} / y) y^{'} ⟹ y^{''} = \frac{( y ^{'} ) ^{2}}{y} . y y^{''} = (y^{'})^{2} .

Try it yourself before checking the explanation above.

The differential equation which represents the family of curves y = c_{1} e^{c_{2} x}, where c_{1} and c_{2} are arbitrary constants is:

(y^{'})^{2} = y

y^{''} = y^{'} y

y y^{''} = y^{'}

y y^{''} = (y^{'})^{2}

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