If the curve $y = c_{1} e^{m_{1} x} + c_{2} e^{m_{3} x}$ , where $c_{1}, c_{2}, c_{3}$ are arbitrary constants and $m_{1}, m_{2}, m_{3}$ are roots of $m^{3} - 7 m + 6 = 0$ , then the differential equation corresponding to the given curve is,

y_{2} - 7 y_{1} + 6 y = 0

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y_{2} - 7 y_{1} + 6 = 0

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y_{2} + 7 y_{1} + 6 y = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

y_{2} + 7 y_{1} + 6 = 0

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Solution

The correct option is A $y_{2} - 7 y_{1} + 6 y = 0$
$\frac{d y}{d x} = m_{1} c_{1} e^{m_{1} x} + m_{3} c_{2} e^{m_{3}} x$

$\frac{d y}{d x} = m_{1} (y - c_{2} e m_{3} x) + m_{3} c_{2} e^{m_{3} x}$

$\frac{d y}{d x} = y_{1} = m_{1} y + (m_{3} - m_{1}) C_{2} e^{m_{3} x}$

$\frac{y_{1} - m_{1} y}{m_{3} - m_{1}} = C_{2} e^{m_{3} x}$

$\frac{d^{2} y}{d x^{2}} = C_{1} m_{1}^{2} e^{m_{1} x} + C_{2} m_{3}^{2} e^{m_{3} x}$

$m_{1}^{2} C y - C_{2} e^{m_{3} x} + C_{2} m_{3}^{2} e^{m_{3} x}$

$y m_{1}^{2} + (m_{3}^{2} - m_{1}^{2}) (C_{2} e^{m_{3} x})$

$y_{2} = y m_{1}^{2} + (m_{3}^{2} - m_{1}^{2}) (\frac{y_{1} m_{1} y}{m_{3} - m_{1}})$

$y_{2} = y m_{1}^{2} + (m_{3} + m_{1}) (y_{1} - m_{1} y)$

$y_{2} = (m_{3} + m_{1}) y_{1} - m_{1} m_{3} y$

$y_{2} - (m_{3} + m_{1}) y_{1} + m_{1} m_{3} y = 0$

$y_{2} - 7 y_{1} + 6 y = 0$

Hence option $A$ is the answer.

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Similar questions

y = c_{1} e^{m_{1} x} + c_{2} e^{m_{3} x}

c_{1}, c_{2}, c_{3}

m_{1}, m_{2}, m_{3}

m^{3} - 7 m + 6 = 0

y_{2} - 7 y_{1} + 12 y = 0

y_{2} - 7 y_{1} + 12 y = 0

y (log 4)

y_{2} - 7 y_{1} + 12 y = 0, y (0) = 2, y_{1} (0) = 7

C_{1} : y^{2} = 4 x, C_{2} : x^{2} + y^{2} - 6 x + 1 = 0

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