What is the solution for y^6y^'= 1/2 (x+1)^{5/2} ?

The solution for y^6y^'= 1/2 (x+1)^{5/2} is y=\sqrt[7]{(x+1)^{7/2}+c_{1}}

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y^6y^'= 1/2 (x+1)^{5/2}

y^{6} y^{^{'}} = \frac{1}{2} (x + 1)^{\frac{5}{2}}

y = 7 (x + 1)^{\frac{7}{2}} + c_{1}

Solution steps

y^{6} y^{'} (x) = \frac{1}{2} (x + 1)^{\frac{5}{2}}

Solve separable ODE:

y = 7 (x + 1)^{\frac{7}{2}} + c_{1}

y = 7 (x + 1)^{\frac{7}{2}} + c_{1}

7 y^{^{'}} + y = 0

f^{^{'}} (x) = - \frac{1}{f ( x )}, f^{^{''}} (x) = - f (x)

\frac{d}{d x} ((y)^{2}) = y^{2} x

\frac{d y}{d x} = \frac{x ^{3} + 1}{y}, y = 2, x = 1

\frac{d y}{d x} y - 6 x^{3} + cos (x) = 0

What is the solution for y^6y^'= 1/2 (x+1)^{5/2} ?
The solution for y^6y^'= 1/2 (x+1)^{5/2} is y=\sqrt[7]{(x+1)^{7/2}+c_{1}}