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1)
![sin2x*(4y^3+4)+y^2y'=0 sin2x*(4y^3+4)+y^2y'=0](https://tex.z-dn.net/?f=sin2x%2A%284y%5E3%2B4%29%2By%5E2y%27%3D0)
![frac{y^2dy}{dx} =-sin2x*(4y^3+4) frac{y^2dy}{dx} =-sin2x*(4y^3+4)](https://tex.z-dn.net/?f=frac%7By%5E2dy%7D%7Bdx%7D+%3D-sin2x%2A%284y%5E3%2B4%29)
![frac{y^2dy}{(4y^3+4)} =-sin2x *dx frac{y^2dy}{(4y^3+4)} =-sin2x *dx](https://tex.z-dn.net/?f=frac%7By%5E2dy%7D%7B%284y%5E3%2B4%29%7D+%3D-sin2x+%2Adx)
![intfrac{y^2dy}{(4y^3+4)} =int-sin2x *dx intfrac{y^2dy}{(4y^3+4)} =int-sin2x *dx](https://tex.z-dn.net/?f=intfrac%7By%5E2dy%7D%7B%284y%5E3%2B4%29%7D+%3Dint-sin2x+%2Adx)
![frac{1}{3} intfrac{3y^2dy}{(4y^3+4)} =int-sin2x *dx frac{1}{3} intfrac{3y^2dy}{(4y^3+4)} =int-sin2x *dx](https://tex.z-dn.net/?f=+frac%7B1%7D%7B3%7D+intfrac%7B3y%5E2dy%7D%7B%284y%5E3%2B4%29%7D+%3Dint-sin2x+%2Adx)
![frac{1}{12} intfrac{d(4y^3+4)}{(4y^3+4)} = frac{1}{2} int-sin2x *d(2x) frac{1}{12} intfrac{d(4y^3+4)}{(4y^3+4)} = frac{1}{2} int-sin2x *d(2x)](https://tex.z-dn.net/?f=+frac%7B1%7D%7B12%7D+intfrac%7Bd%284y%5E3%2B4%29%7D%7B%284y%5E3%2B4%29%7D+%3D+frac%7B1%7D%7B2%7D+int-sin2x+%2Ad%282x%29)
![frac{1}{12}ln(4y^3+4)= frac{1}{2} cos 2x+C frac{1}{12}ln(4y^3+4)= frac{1}{2} cos 2x+C](https://tex.z-dn.net/?f=+frac%7B1%7D%7B12%7Dln%284y%5E3%2B4%29%3D+frac%7B1%7D%7B2%7D+cos+2x%2BC)
![ln(4y^3+4)= 6cos 2x+12C ln(4y^3+4)= 6cos 2x+12C](https://tex.z-dn.net/?f=ln%284y%5E3%2B4%29%3D+6cos+2x%2B12C)
![4y^3+4= e^{6cos 2x}*e^{12C} 4y^3+4= e^{6cos 2x}*e^{12C}](https://tex.z-dn.net/?f=4y%5E3%2B4%3D+e%5E%7B6cos+2x%7D%2Ae%5E%7B12C%7D)
![y=sqrt[3]{frac{Ce^{6cos 2x}-4}{4}} y=sqrt[3]{frac{Ce^{6cos 2x}-4}{4}}](https://tex.z-dn.net/?f=y%3Dsqrt%5B3%5D%7Bfrac%7BCe%5E%7B6cos+2x%7D-4%7D%7B4%7D%7D)
![1=sqrt[3]{frac{Ce^{6cos 2*0}-4}{4}} 1=sqrt[3]{frac{Ce^{6cos 2*0}-4}{4}}](https://tex.z-dn.net/?f=1%3Dsqrt%5B3%5D%7Bfrac%7BCe%5E%7B6cos+2%2A0%7D-4%7D%7B4%7D%7D)
![8=Ce^{6} 8=Ce^{6}](https://tex.z-dn.net/?f=8%3DCe%5E%7B6%7D)
![C=8e^{-6} C=8e^{-6}](https://tex.z-dn.net/?f=C%3D8e%5E%7B-6%7D)
![y=sqrt[3]{frac{8e^{-6}*e^{6cos 2x}-4}{4}} y=sqrt[3]{frac{8e^{-6}*e^{6cos 2x}-4}{4}}](https://tex.z-dn.net/?f=y%3Dsqrt%5B3%5D%7Bfrac%7B8e%5E%7B-6%7D%2Ae%5E%7B6cos+2x%7D-4%7D%7B4%7D%7D)
2)
![y'-5y=x^4e^{5x} y'-5y=x^4e^{5x}](https://tex.z-dn.net/?f=y%27-5y%3Dx%5E4e%5E%7B5x%7D)
![y=uv y=uv](https://tex.z-dn.net/?f=y%3Duv)
![u'v+uv'-5uv=x^4e^{5x} u'v+uv'-5uv=x^4e^{5x}](https://tex.z-dn.net/?f=u%27v%2Buv%27-5uv%3Dx%5E4e%5E%7B5x%7D)
![left { {{v'-5v=0} atop {u'v=x^4e^{5x}}} right. left { {{v'-5v=0} atop {u'v=x^4e^{5x}}} right.](https://tex.z-dn.net/?f=+left+%7B+%7B%7Bv%27-5v%3D0%7D+atop+%7Bu%27v%3Dx%5E4e%5E%7B5x%7D%7D%7D+right.+)
![frac{dv}{dx}=5v frac{dv}{dx}=5v](https://tex.z-dn.net/?f=+frac%7Bdv%7D%7Bdx%7D%3D5v+)
![lnv=5x lnv=5x](https://tex.z-dn.net/?f=lnv%3D5x)
![v=e^{5x} v=e^{5x}](https://tex.z-dn.net/?f=v%3De%5E%7B5x%7D)
![u'e^{5x}=x^4e^{5x} u'e^{5x}=x^4e^{5x}](https://tex.z-dn.net/?f=u%27e%5E%7B5x%7D%3Dx%5E4e%5E%7B5x%7D)
![frac{du}{dx} =x^4 frac{du}{dx} =x^4](https://tex.z-dn.net/?f=+frac%7Bdu%7D%7Bdx%7D+%3Dx%5E4)
![u= frac{x^5}{5} +C u= frac{x^5}{5} +C](https://tex.z-dn.net/?f=u%3D+frac%7Bx%5E5%7D%7B5%7D+%2BC)
![y=uv=e^{5x}(frac{x^5}{5} +C) y=uv=e^{5x}(frac{x^5}{5} +C)](https://tex.z-dn.net/?f=y%3Duv%3De%5E%7B5x%7D%28frac%7Bx%5E5%7D%7B5%7D+%2BC%29)
3)
![2y''+5y'=0 2y''+5y'=0](https://tex.z-dn.net/?f=2y%27%27%2B5y%27%3D0)
![y=e^{kx} y=e^{kx}](https://tex.z-dn.net/?f=y%3De%5E%7Bkx%7D)
![2k^2e^{kx}+5ke^{kx}=0 2k^2e^{kx}+5ke^{kx}=0](https://tex.z-dn.net/?f=2k%5E2e%5E%7Bkx%7D%2B5ke%5E%7Bkx%7D%3D0)
![e^{kx}(2k^2+5k)=0 e^{kx}(2k^2+5k)=0](https://tex.z-dn.net/?f=e%5E%7Bkx%7D%282k%5E2%2B5k%29%3D0)
![k(2k+5)=0 k(2k+5)=0](https://tex.z-dn.net/?f=k%282k%2B5%29%3D0)
![k_1=0;k_2=-2.5 k_1=0;k_2=-2.5](https://tex.z-dn.net/?f=k_1%3D0%3Bk_2%3D-2.5)
![y=C_1+C_2e^{-2.5x} y=C_1+C_2e^{-2.5x}](https://tex.z-dn.net/?f=y%3DC_1%2BC_2e%5E%7B-2.5x%7D)
2)
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