Which of the following differential equations has y - c 1 e x - c 2 e - x as the general solution-a d 2 y - dx 2- y -0b d 2 y - d x 2- y -0c d 2 y - d x 2-1-0d d 2 y - dx 2-1-0

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General Solution of a Differential Equation

Which of the ...

Question

Which of the following differential equations has $y = c_{1} e^{x} + c_{2} e^{- x}$ as the general solution?
(a) $\frac{d^{2} y}{d x^{2}} + y = 0$
(b) $\frac{d^{2} y}{d x^{2}} - y = 0$
(c) $\frac{d^{2} y}{d x^{2}} + 1 = 0$
(d) $\frac{d^{2} y}{d x^{2}} - 1 = 0$

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Solution

Given, general solution is $y = c_{1} e^{x} + c_{2} e^{- x} . . . (i)$
On differentiating twice w.r.t. x, we get $y^{'} = c_{1} e^{x} + c_{2} e^{- x} (- 1)$
Again, differentiating w.r.t. x, we get
$y^{''} = c_{1} e^{x} - c_{2} e^{- x} (- 1) \Rightarrow y^{''} = c_{1} e^{x} + c_{2} e^{- x} \Rightarrow y^{''} = y \Rightarrow y^{''} - y = 0$
Which is the required differential equation of the given general solution. Replacing y' by $\frac{d y}{d x}$ and y'' by $\frac{d^{2} y}{d^{x^{2}}} .$ Hence, option (b) is correct option.

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[MP PET 1993]

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Which of the following differential equations has y=c1ex+c2e−x as the general solution? (a) d2ydx2+y=0 (b) d2ydx2−y=0 (c) d2ydx2+1=0 (d) d2ydx2−1=0

Which of the following differential equations has $y = c_{1} e^{x} + c_{2} e^{- x}$ as the general solution?
(a) $\frac{d^{2} y}{d x^{2}} + y = 0$
(b) $\frac{d^{2} y}{d x^{2}} - y = 0$
(c) $\frac{d^{2} y}{d x^{2}} + 1 = 0$
(d) $\frac{d^{2} y}{d x^{2}} - 1 = 0$