Всем доброго вечера :) Помогите пожалуйста с дифференциальным уравнением. Определить его тип и решить :) Спасибо!!!
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Уравнение Бернулли.
![y'+ frac{1}{x} cdot y= frac{lnx}{x} cdot y^2\\y=uv; ,; ; y'=u'v+uv'\\u'v+uv'+frac{uv}{x}= frac{lnx}{x} cdot (uv)^2\\u'v+u(v'+frac{v}{x})= frac{lnx}{x}cdot u^2v^2\\a); ; v'+frac{v}{x}=0; ,; ; frac{dv}{dx}=-frac{v}{x} ; ,; ; int frac{dv}{v}=-int frac{dx}{x}\\lnv=-lnx; ,; ; lmv=ln(x^{-1}); ,; ; v=frac{1}{x} y'+ frac{1}{x} cdot y= frac{lnx}{x} cdot y^2\\y=uv; ,; ; y'=u'v+uv'\\u'v+uv'+frac{uv}{x}= frac{lnx}{x} cdot (uv)^2\\u'v+u(v'+frac{v}{x})= frac{lnx}{x}cdot u^2v^2\\a); ; v'+frac{v}{x}=0; ,; ; frac{dv}{dx}=-frac{v}{x} ; ,; ; int frac{dv}{v}=-int frac{dx}{x}\\lnv=-lnx; ,; ; lmv=ln(x^{-1}); ,; ; v=frac{1}{x}](https://tex.z-dn.net/?f=y%27%2B+frac%7B1%7D%7Bx%7D+cdot+y%3D+frac%7Blnx%7D%7Bx%7D+cdot+y%5E2%5C%5Cy%3Duv%3B+%2C%3B+%3B+y%27%3Du%27v%2Buv%27%5C%5Cu%27v%2Buv%27%2Bfrac%7Buv%7D%7Bx%7D%3D+frac%7Blnx%7D%7Bx%7D+cdot+%28uv%29%5E2%5C%5Cu%27v%2Bu%28v%27%2Bfrac%7Bv%7D%7Bx%7D%29%3D+frac%7Blnx%7D%7Bx%7Dcdot+u%5E2v%5E2%5C%5Ca%29%3B+%3B+v%27%2Bfrac%7Bv%7D%7Bx%7D%3D0%3B+%2C%3B+%3B+frac%7Bdv%7D%7Bdx%7D%3D-frac%7Bv%7D%7Bx%7D+%3B+%2C%3B+%3B+int++frac%7Bdv%7D%7Bv%7D%3D-int++frac%7Bdx%7D%7Bx%7D%5C%5Clnv%3D-lnx%3B+%2C%3B+%3B+lmv%3Dln%28x%5E%7B-1%7D%29%3B+%2C%3B+%3B+v%3Dfrac%7B1%7D%7Bx%7D)
![b); ; u'cdot frac{1}{x} = frac{lnx}{x}cdot u^2cdot frac{1}{x^2}\\ frac{du}{dx} = frac{lnx}{x^2} cdot u^2\\int frac{du}{u^2}=int frac{lnx}{x^2} b); ; u'cdot frac{1}{x} = frac{lnx}{x}cdot u^2cdot frac{1}{x^2}\\ frac{du}{dx} = frac{lnx}{x^2} cdot u^2\\int frac{du}{u^2}=int frac{lnx}{x^2}](https://tex.z-dn.net/?f=b%29%3B+%3B+u%27cdot++frac%7B1%7D%7Bx%7D+%3D+frac%7Blnx%7D%7Bx%7Dcdot+u%5E2cdot+frac%7B1%7D%7Bx%5E2%7D%5C%5C+frac%7Bdu%7D%7Bdx%7D+%3D+frac%7Blnx%7D%7Bx%5E2%7D+cdot+u%5E2%5C%5Cint+frac%7Bdu%7D%7Bu%5E2%7D%3Dint+frac%7Blnx%7D%7Bx%5E2%7D)
![int u^{-2}cdot du=int x^{-2}cdot lnxcdot dx\\ Big [u=lnx,; dv=x^{-2}dx,; v=frac{x^{-1}}{-1}=-frac{1}{x},du=frac{dx}{x},; int u, dv=uv-int v, duBig ]\\frac{u^{-1}}{-1}=lnxcdot (-frac{1}{x})+int frac{dx}{x^2}\\-frac{1}{u}=-frac{lnx}{x}-frac{1}{x}-C; ; ; Rightarrow ; ; ; frac{1}{u}=frac{lnx}{x} +frac{1}{x}+C\\ frac{1}{u}= frac{lnx+1+Cx}{x} ; ; ; Rightarrow ; ; ; u= frac{x}{lnx+1+Cx} int u^{-2}cdot du=int x^{-2}cdot lnxcdot dx\\ Big [u=lnx,; dv=x^{-2}dx,; v=frac{x^{-1}}{-1}=-frac{1}{x},du=frac{dx}{x},; int u, dv=uv-int v, duBig ]\\frac{u^{-1}}{-1}=lnxcdot (-frac{1}{x})+int frac{dx}{x^2}\\-frac{1}{u}=-frac{lnx}{x}-frac{1}{x}-C; ; ; Rightarrow ; ; ; frac{1}{u}=frac{lnx}{x} +frac{1}{x}+C\\ frac{1}{u}= frac{lnx+1+Cx}{x} ; ; ; Rightarrow ; ; ; u= frac{x}{lnx+1+Cx}](https://tex.z-dn.net/?f=int+u%5E%7B-2%7Dcdot+du%3Dint+x%5E%7B-2%7Dcdot+lnxcdot+dx%5C%5C+Big+%5Bu%3Dlnx%2C%3B+dv%3Dx%5E%7B-2%7Ddx%2C%3B+v%3Dfrac%7Bx%5E%7B-1%7D%7D%7B-1%7D%3D-frac%7B1%7D%7Bx%7D%2Cdu%3Dfrac%7Bdx%7D%7Bx%7D%2C%3B+int+u%2C+dv%3Duv-int+v%2C+duBig+%5D%5C%5Cfrac%7Bu%5E%7B-1%7D%7D%7B-1%7D%3Dlnxcdot+%28-frac%7B1%7D%7Bx%7D%29%2Bint+frac%7Bdx%7D%7Bx%5E2%7D%5C%5C-frac%7B1%7D%7Bu%7D%3D-frac%7Blnx%7D%7Bx%7D-frac%7B1%7D%7Bx%7D-C%3B+%3B+%3B+Rightarrow+%3B+%3B+%3B++frac%7B1%7D%7Bu%7D%3Dfrac%7Blnx%7D%7Bx%7D+%2Bfrac%7B1%7D%7Bx%7D%2BC%5C%5C++frac%7B1%7D%7Bu%7D%3D+frac%7Blnx%2B1%2BCx%7D%7Bx%7D+%3B+%3B+%3B+Rightarrow+%3B+%3B+%3B+u%3D+frac%7Bx%7D%7Blnx%2B1%2BCx%7D)
![c); ; y=uv=frac{1}{x}cdot frac{x}{lnx+1+Cx} \\y=frac{1}{lnx+Cx+1} c); ; y=uv=frac{1}{x}cdot frac{x}{lnx+1+Cx} \\y=frac{1}{lnx+Cx+1}](https://tex.z-dn.net/?f=c%29%3B+%3B+y%3Duv%3Dfrac%7B1%7D%7Bx%7Dcdot++frac%7Bx%7D%7Blnx%2B1%2BCx%7D+%5C%5Cy%3Dfrac%7B1%7D%7Blnx%2BCx%2B1%7D)
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спасибо огромное!)
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