Sum

Obtain the differential equation by eliminating arbitrary constants from the following equations.

y = (c_{1} + c_{2} x) e^{x}

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#### Solution

y = (c_{1} + c_{2} x) e^{x}

∴ ye ^{-x} = c_{1 }+ c_{2}x

Differentiating w.r.t. x, we get

`y (-e^-x) + e ^-x dy/dx = 0 + c_2`

∴`e^-x(dy/dx - y) = c_2`

Again, differentiating w.r.t. x, we get

`e^-x((d^2y)/dx^2-dy/dx) - e^-x(dy/dx-y) = 0`

∴`e^-x((d^2y)/dx^2-dy/dx - dy/dx+ y) = 0`

∴`(d^2y)/dx^2 - 2dy/dx + y = 0`

Concept: Formation of Differential Equation by Eliminating Arbitary Constant

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