What is the solution for y^2e^x+3ye^{2x}(dy)/(dx)=2e^x,y(0)=1 ?

The solution for y^2e^x+3ye^{2x}(dy)/(dx)=2e^x,y(0)=1 is y=sqrt(-e^{2/(3e^x)-2/3)+2}

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y^2e^x+3ye^{2x}(dy)/(dx)=2e^x,y(0)=1

y^{2} e^{x} + 3 y e^{2 x} \frac{d y}{d x} = 2 e^{x}, y (0) = 1

y = - e^{\frac{2}{3 e ^{x}} - \frac{2}{3}} + 2

Solution steps

y^{2} e^{x} + 3 y e^{2 x} \frac{d y}{d x} = 2 e^{x}

Solve separable ODE:

y = - e^{\frac{2}{3 e ^{x}} - \frac{2}{3}} + 2

y = - e^{\frac{2}{3 e ^{x}} - \frac{2}{3}} + 2

\frac{x ^{2}}{y ^{2} - 1} y^{^{'}} = \frac{3 x ^{3}}{y}

y^{^{'}} = 9 y

(x + 1) \frac{d y}{d x} + y = 10

x^{^{'}} + x = - 1

y^{^{'}} = \frac{x y sin ( x ^{2} )}{y + 1}

What is the solution for y^2e^x+3ye^{2x}(dy)/(dx)=2e^x,y(0)=1 ?
The solution for y^2e^x+3ye^{2x}(dy)/(dx)=2e^x,y(0)=1 is y=sqrt(-e^{2/(3e^x)-2/3)+2}