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Find the integrating factor of the differential equation. ((e^(-2^√x))/(√x)-y/(√x))dy/dx=1 - Mathematics

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Find the integrating factor of the differential equation.

`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`

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Solution

We have

`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`

`dy/dx=(e^(-2^sqrtx))/sqrtx-y/sqrtx`

`=>dy/dx+(1/sqrtx)y=(e^(-2^sqrtx))/sqrtx`

It is in the form dy/dx+Py=Q, where P and Q are the constants or functions of x.
Thus, the integrating factor of the given differential equation is

`e^(intPdx)=e^(int1/sqrtxdx)`

`=e^(intx^(-1/2)dx)=e^(x^((1/2))/(1/2))`

`=e^(2sqrtx)`

Concept: Solutions of Linear Differential Equation
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