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Дифференциальное уравнение Бернулли:
![y`+y=e^frac{x}{2}y^frac{1}{2}\frac{y`}{y^frac{1}{2}}+y^frac{1}{2}=e^frac{x}{2}\\z=y^frac{1}{2}= textgreater z`=frac{y`}{2sqrt y}\frac{y`}{sqrt y}=2z`\2z`+z=e^frac{x}{2}\z=uv ;z`=u`v+v`u\2u`v+2v`u+uv=e^frac{x}{2}\2u`v+u(2v`+v)=e^frac{x}{2}\begin{cases}2v`+v=0\2u`v=e^frac{x}{2}end{cases} y`+y=e^frac{x}{2}y^frac{1}{2}\frac{y`}{y^frac{1}{2}}+y^frac{1}{2}=e^frac{x}{2}\\z=y^frac{1}{2}= textgreater z`=frac{y`}{2sqrt y}\frac{y`}{sqrt y}=2z`\2z`+z=e^frac{x}{2}\z=uv ;z`=u`v+v`u\2u`v+2v`u+uv=e^frac{x}{2}\2u`v+u(2v`+v)=e^frac{x}{2}\begin{cases}2v`+v=0\2u`v=e^frac{x}{2}end{cases}](https://tex.z-dn.net/?f=y%60%2By%3De%5Efrac%7Bx%7D%7B2%7Dy%5Efrac%7B1%7D%7B2%7D%5Cfrac%7By%60%7D%7By%5Efrac%7B1%7D%7B2%7D%7D%2By%5Efrac%7B1%7D%7B2%7D%3De%5Efrac%7Bx%7D%7B2%7D%5C%5Cz%3Dy%5Efrac%7B1%7D%7B2%7D%3D+textgreater++z%60%3Dfrac%7By%60%7D%7B2sqrt+y%7D%5Cfrac%7By%60%7D%7Bsqrt+y%7D%3D2z%60%5C2z%60%2Bz%3De%5Efrac%7Bx%7D%7B2%7D%5Cz%3Duv+%3Bz%60%3Du%60v%2Bv%60u%5C2u%60v%2B2v%60u%2Buv%3De%5Efrac%7Bx%7D%7B2%7D%5C2u%60v%2Bu%282v%60%2Bv%29%3De%5Efrac%7Bx%7D%7B2%7D%5Cbegin%7Bcases%7D2v%60%2Bv%3D0%5C2u%60v%3De%5Efrac%7Bx%7D%7B2%7Dend%7Bcases%7D)
![2frac{dv}{dx}=-v\frac{dv}{v}=-frac{dx}{2}\intfrac{dv}{v}=-frac{1}{2}int dx\ln|v|=-frac{x}{2}\v=e^{-frac{x}{2}}=frac{1}{e^frac{x}{2}}\frac{2u`}{e^frac{x}{2}}=e^{frac{x}{2}}\u`=frac{e^x}{2}\u=frac{1}{2}int e^xdx=frac{e^x}{2}+C\z=uv=(frac{e^x}{2}+C)*frac{1}{e^frac{x}{2}}=frac{e^frac{x}{2}}{2}+frac{C}{e^frac{x}{2}}\z=sqrt y= textgreater y=z^2\y=(frac{e^frac{x}{2}}{2}+frac{C}{e^frac{x}{2}})^2=frac{e^x}{4}+C+frac{C^2}{e^x}\OTBET:y=frac{e^x}{4}+frac{C^2}{e^x}+C;C-const;y=0 2frac{dv}{dx}=-v\frac{dv}{v}=-frac{dx}{2}\intfrac{dv}{v}=-frac{1}{2}int dx\ln|v|=-frac{x}{2}\v=e^{-frac{x}{2}}=frac{1}{e^frac{x}{2}}\frac{2u`}{e^frac{x}{2}}=e^{frac{x}{2}}\u`=frac{e^x}{2}\u=frac{1}{2}int e^xdx=frac{e^x}{2}+C\z=uv=(frac{e^x}{2}+C)*frac{1}{e^frac{x}{2}}=frac{e^frac{x}{2}}{2}+frac{C}{e^frac{x}{2}}\z=sqrt y= textgreater y=z^2\y=(frac{e^frac{x}{2}}{2}+frac{C}{e^frac{x}{2}})^2=frac{e^x}{4}+C+frac{C^2}{e^x}\OTBET:y=frac{e^x}{4}+frac{C^2}{e^x}+C;C-const;y=0](https://tex.z-dn.net/?f=2frac%7Bdv%7D%7Bdx%7D%3D-v%5Cfrac%7Bdv%7D%7Bv%7D%3D-frac%7Bdx%7D%7B2%7D%5Cintfrac%7Bdv%7D%7Bv%7D%3D-frac%7B1%7D%7B2%7Dint+dx%5Cln%7Cv%7C%3D-frac%7Bx%7D%7B2%7D%5Cv%3De%5E%7B-frac%7Bx%7D%7B2%7D%7D%3Dfrac%7B1%7D%7Be%5Efrac%7Bx%7D%7B2%7D%7D%5Cfrac%7B2u%60%7D%7Be%5Efrac%7Bx%7D%7B2%7D%7D%3De%5E%7Bfrac%7Bx%7D%7B2%7D%7D%5Cu%60%3Dfrac%7Be%5Ex%7D%7B2%7D%5Cu%3Dfrac%7B1%7D%7B2%7Dint+e%5Exdx%3Dfrac%7Be%5Ex%7D%7B2%7D%2BC%5Cz%3Duv%3D%28frac%7Be%5Ex%7D%7B2%7D%2BC%29%2Afrac%7B1%7D%7Be%5Efrac%7Bx%7D%7B2%7D%7D%3Dfrac%7Be%5Efrac%7Bx%7D%7B2%7D%7D%7B2%7D%2Bfrac%7BC%7D%7Be%5Efrac%7Bx%7D%7B2%7D%7D%5Cz%3Dsqrt+y%3D+textgreater++y%3Dz%5E2%5Cy%3D%28frac%7Be%5Efrac%7Bx%7D%7B2%7D%7D%7B2%7D%2Bfrac%7BC%7D%7Be%5Efrac%7Bx%7D%7B2%7D%7D%29%5E2%3Dfrac%7Be%5Ex%7D%7B4%7D%2BC%2Bfrac%7BC%5E2%7D%7Be%5Ex%7D%5COTBET%3Ay%3Dfrac%7Be%5Ex%7D%7B4%7D%2Bfrac%7BC%5E2%7D%7Be%5Ex%7D%2BC%3BC-const%3By%3D0)
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