Internal problem ID [15103]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 8. First order not solved for the derivative. Exercises page 67
Problem number: 215.
ODE order: 1.
ODE degree: 6.
CAS Maple gives this as type [_quadrature]
\[ \boxed {x \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=a} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 233
dsolve(x*(1+diff(y(x),x)^2)^(3/2)=a,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \int \frac {\sqrt {\left (a \,x^{2}\right )^{\frac {2}{3}}-x^{2}}}{x}d x +c_{1} \\ y \left (x \right ) &= -\frac {\left (\int \frac {\sqrt {-2 i \sqrt {3}\, \left (a \,x^{2}\right )^{\frac {2}{3}}-2 \left (a \,x^{2}\right )^{\frac {2}{3}}-4 x^{2}}}{x}d x \right )}{2}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\sqrt {-2 i \sqrt {3}\, \left (a \,x^{2}\right )^{\frac {2}{3}}-2 \left (a \,x^{2}\right )^{\frac {2}{3}}-4 x^{2}}}{x}d x \right )}{2}+c_{1} \\ y \left (x \right ) &= -\left (\int \frac {\sqrt {\left (a \,x^{2}\right )^{\frac {2}{3}}-x^{2}}}{x}d x \right )+c_{1} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \left (\int \frac {\sqrt {i \sqrt {3}\, \left (a \,x^{2}\right )^{\frac {2}{3}}-\left (a \,x^{2}\right )^{\frac {2}{3}}-2 x^{2}}}{x}d x \right )}{2}+c_{1} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \left (\int \frac {\sqrt {i \sqrt {3}\, \left (a \,x^{2}\right )^{\frac {2}{3}}-\left (a \,x^{2}\right )^{\frac {2}{3}}-2 x^{2}}}{x}d x \right )}{2}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 19.313 (sec). Leaf size: 375
DSolve[x*(1+y'[x]^2)^(3/2)==a,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt [3]{x} \sqrt {\frac {a^{2/3}}{x^{2/3}}-1} \left (x^{2/3}-a^{2/3}\right )+c_1 \\ y(x)\to \sqrt [3]{x} \sqrt {\frac {a^{2/3}}{x^{2/3}}-1} \left (a^{2/3}-x^{2/3}\right )+c_1 \\ y(x)\to c_1-\frac {1}{2} \sqrt [3]{x} \sqrt {-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}} \left (2 x^{2/3}+\left (1-i \sqrt {3}\right ) a^{2/3}\right ) \\ y(x)\to \frac {1}{2} \sqrt [3]{x} \sqrt {-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}} \left (2 x^{2/3}+\left (1-i \sqrt {3}\right ) a^{2/3}\right )+c_1 \\ y(x)\to c_1-\frac {1}{2} \sqrt [3]{x} \sqrt {-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}} \left (2 x^{2/3}+\left (1+i \sqrt {3}\right ) a^{2/3}\right ) \\ y(x)\to \frac {1}{2} \sqrt [3]{x} \sqrt {-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}} \left (2 x^{2/3}+\left (1+i \sqrt {3}\right ) a^{2/3}\right )+c_1 \\ \end{align*}