What is the solution for (dy)/(dt)=(1+4t)sqrt(y) ?

The solution for (dy)/(dt)=(1+4t)sqrt(y) is y=(t^2)/4+(c_{1}t)/2+(c_{1}^2)/4+t^3+c_{1}t^2+t^4

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(dy)/(dt)=(1+4t)sqrt(y)

\frac{d y}{d t} = (1 + 4 t) y

y = \frac{t ^{2}}{4} + \frac{c _{1} t}{2} + \frac{c _{1}^{2}}{4} + t^{3} + c_{1} t^{2} + t^{4}

Solution steps

\frac{d y}{d t} = (1 + 4 t) y

Solve separable ODE:

y = \frac{t ^{2}}{4} + \frac{c _{1} t}{2} + \frac{c _{1}^{2}}{4} + t^{3} + c_{1} t^{2} + t^{4}

y = \frac{t ^{2}}{4} + \frac{c _{1} t}{2} + \frac{c _{1}^{2}}{4} + t^{3} + c_{1} t^{2} + t^{4}

\frac{d y}{d x} + 2 x y = x, y (0) = 2

y^{^{'}} = 3 x^{2} y^{2}

\frac{d y}{d t} = \frac{t}{y - t ^{2} y}, y (0) = 4

x \frac{d y}{d x} = e^{6 y + x}, x > 0

\frac{d N}{d t} = k N (t)

What is the solution for (dy)/(dt)=(1+4t)sqrt(y) ?
The solution for (dy)/(dt)=(1+4t)sqrt(y) is y=(t^2)/4+(c_{1}t)/2+(c_{1}^2)/4+t^3+c_{1}t^2+t^4