Дифференциальные урвнения

1) [latex]sin2x*(4y^3+4)+y^2y'=0[/latex] [latex]\frac{y^2dy}{dx} =-sin2x*(4y^3+4)[/latex] [latex]\frac{y^2dy}{(4y^3+4)} =-sin2x *dx[/latex] [latex]\int\frac{y^2dy}{(4y^3+4)} =\int-sin2x *dx[/latex] [latex] \frac{1}{3} \int\frac{3y^2dy}{(4y^3+4)} =\int-sin2x *dx[/latex] [latex] \frac{1}{12} \int\frac{d(4y^3+4)}{(4y^3+4)} = \frac{1}{2} \int-sin2x *d(2x)[/latex] [latex] \frac{1}{12}ln(4y^3+4)= \frac{1}{2} cos 2x+C[/latex] [latex]ln(4y^3+4)= 6cos 2x+12C[/latex] [latex]4y^3+4= e^{6cos 2x}*e^{12C}[/latex] [latex]y=\sqrt[3]{\frac{Ce^{6cos 2x}-4}{4}}[/latex] [latex]1=\sqrt[3]{\frac{Ce^{6cos 2*0}-4}{4}}[/latex] [latex]8=Ce^{6}[/latex] [latex]C=8e^{-6}[/latex] [latex]y=\sqrt[3]{\frac{8e^{-6}*e^{6cos 2x}-4}{4}}[/latex] 2) [latex]y'-5y=x^4e^{5x}[/latex] [latex]y=uv[/latex] [latex]u'v+uv'-5uv=x^4e^{5x}[/latex] [latex] \left \{ {{v'-5v=0} \atop {u'v=x^4e^{5x}}} \right. [/latex] [latex] \frac{dv}{dx}=5v [/latex] [latex]lnv=5x[/latex] [latex]v=e^{5x}[/latex] [latex]u'e^{5x}=x^4e^{5x}[/latex] [latex] \frac{du}{dx} =x^4[/latex] [latex]u= \frac{x^5}{5} +C[/latex] [latex]y=uv=e^{5x}(\frac{x^5}{5} +C)[/latex] 3) [latex]2y''+5y'=0[/latex] [latex]y=e^{kx}[/latex] [latex]2k^2e^{kx}+5ke^{kx}=0[/latex] [latex]e^{kx}(2k^2+5k)=0[/latex] [latex]k(2k+5)=0[/latex] [latex]k_1=0;k_2=-2.5[/latex] [latex]y=C_1+C_2e^{-2.5x}[/latex]