The solution of y 2-7 y 1-12 y -0 isA- None of these B- y - C 1 x e 3 x - C 2 e 4 xC- y - C 1 e 3 x - C 2 xe 4 xD- y - C 1 e 3 x - C 2 e 4 x

The correct option is D y = C_1 e^3x + C_2 e^4xThe given equation can be written as (d/dx 3)(dy/dx 4y) = 0 If dy/dx 4y = u then (I) reduces to du/dx 3u = 0 ...

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Standard XII

Mathematics

Solving Linear Differential Equations of First Order

The solution ...

Question

The solution of $y_{2} - 7 y_{1} + 12 y = 0$ is

y = C_{1} e^{3 x} + C_{2} e^{4 x}

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y = C_{1} x e^{3 x} + C_{2} e^{4 x}

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y = C_{1} e^{3 x} + C_{2} x e^{4 x}

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Solution

The correct option is A $y = C_{1} e^{3 x} + C_{2} e^{4 x}$
The given equation can be written as $(\frac{d}{d x} - 3) (\frac{d y}{d x} - 4 y) = 0$
If $\frac{d y}{d x} - 4 y = u$ then (I) reduces to $\frac{d u}{d x} - 3 u = 0$
$\Rightarrow \frac{d u}{u} = 3 d x \Rightarrow u = C_{1} e^{3 x}$ . Therefore, we have $\frac{d y}{d x} - 4 y = C_{1} e^{3 x}$ which is a linear equation whose I.F. is
$e^{4 x} . S o \frac{d}{d x} (y e^{- 4 x}) = C_{1} e^{- x} \Rightarrow y e^{- 4 x} = - C_{1} e^{- x} + C_{2} \Rightarrow y = C_{1}^{3 x} + C_{2} e^{4 x}$

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The solution of y2−7y1+12y=0 is

The solution of $y_{2} - 7 y_{1} + 12 y = 0$ is