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_{1}x\,

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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