What is the solution for (d^2y)/(dx^2)=sqrt(1+(dy)/(dx)) ?

The solution for (d^2y)/(dx^2)=sqrt(1+(dy)/(dx)) is y= 1/4 ((x^3)/3+c_{1}^2x)+(c_{1}x^2)/4-x+c_{2}

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(d^2y)/(dx^2)=sqrt(1+(dy)/(dx))

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

y = \frac{1}{4} (\frac{x ^{3}}{3} + c_{1}^{2} x) + \frac{c _{1} x ^{2}}{4} - x + c_{2}

Solution steps

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

Solve using substitution:

y = \frac{1}{4} (\frac{x ^{3}}{3} + c_{1}^{2} x) + \frac{c _{1} x ^{2}}{4} - x + c_{2}

y = \frac{1}{4} (\frac{x ^{3}}{3} + c_{1}^{2} x) + \frac{c _{1} x ^{2}}{4} - x + c_{2}

x \to 0 lim (\frac{1}{sin ^{2} ( 2 x )})

y^{^{'}} = t y + t

x y^{^{'}} + y = - 5 x y^{2}

\frac{d}{d x} ((2 x - 4) (- 3 x^{2} + 1))

x \to - 5 lim (\frac{5 - ∣ x ∣}{5 + x})

What is the solution for (d^2y)/(dx^2)=sqrt(1+(dy)/(dx)) ?
The solution for (d^2y)/(dx^2)=sqrt(1+(dy)/(dx)) is y= 1/4 ((x^3)/3+c_{1}^2x)+(c_{1}x^2)/4-x+c_{2}