The differential equation of the family of curves y - Ae 3 x - Be 5 x- where A and B are arbitrary constants- isA- d2 y-d x2-8 d y-d x-15 y-0B- d2 y-d x2-d y-d x-y-0C- d2 y-d x2-8 d y-d x-15 y-0D- N one of these

The correct option is A d^2 y/dx^2 8 dy/dx+15y =0 y=Ae^3x+Be^5x ⇒dy/dx=3 Ae^3x+5Be^5x⇒d^2 y/dx^2=9 Ae^3x+25Be^5x ⇒d^2 y/dx^2 8 dy/dx+15y =0 (By inspection) ...

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Byju's Answer

Standard XII

Mathematics

Formation of a Differential Equation from a General Solution

The different...

Question

The differential equation of the family of curves $y = A e^{3 x} + B e^{5 x}$ ,where A and B are arbitrary constants, is

\frac{d^{2} y}{d x^{2}} + 8 \frac{d y}{d x} + 15 y = 0

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\frac{d^{2} y}{d x^{2}} - 8 \frac{d y}{d x} + 15 y = 0

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\frac{d^{2} y}{d x^{2}} - \frac{d y}{d x} + y = 0

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Solution

The correct option is B $\frac{d^{2} y}{d x^{2}} - 8 \frac{d y}{d x} + 15 y = 0$

$y = A e^{3 x} + B e^{5 x}$
$\Rightarrow \frac{d y}{d x} = 3 A e^{3 x} + 5 B e^{5 x}$
$\Rightarrow \frac{d^{2} y}{d x^{2}} = 9 A e^{3 x} + 25 B e^{5 x}$
By checking all the options, we have
$∴ \frac{d^{2} y}{d x^{2}} - 8 \frac{d y}{d x} + 15 y$
$= 9 A e^{3 x} + 25 B e^{5 x} - 8 (3 A e^{3 x} + 5 B e^{5 x}) + 15 (A e^{3 x} + B e^{5 x})$
$= 9 A e^{3 x} + 25 B e^{5 x} - 24 A e^{3 x} - 40 B e^{5 x} + 15 A e^{3 x} + 15 B e^{5 x}$
$= 9 A e^{3 x} - 24 A e^{3 x} + 15 A e^{3 x} + 25 B e^{5 x} - 40 B e^{5 x} + 15 B e^{5 x}$
$= 0$
$∴ \frac{d^{2} y}{d x^{2}} - 8 \frac{d y}{d x} + 15 y = 0$

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