ODE1.3

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Solve $y' = \frac{y+x}{x},\,\,\,y(1) = 7$

Make the variable transformation $y=vx\,$

$y' = v + x\,\frac{dv}{dx}\,$

$v + x\,\frac{dv}{dx} = \frac{vx+x}{x} = v + 1\,$

$x \, \frac{dv}{dx} = 1\,$

$dv = \frac{dx}{x}\,$

$v(x) = \ln|x| + c_1\,$

$\frac{y(x)}{x} = \ln|x| + c_1\,$

$y(x) = x \ln|x| + c_1x\,$

$y(1) = 1\cdot 0 + c_1 = 7,\,\,\,c_1=7\,$

The unique solution is

$y(x) = x \ln|x| + 7x\,$

Ordinary Differential Equations

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