Internal problem ID [8883]
Internal file name [OUTPUT/7818_Monday_June_06_2022_12_35_54_AM_11688120/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 549.
ODE order: 1.
ODE degree: 6.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {x^{2} \left ({y^{\prime }}^{2}+1\right )^{3}=a^{2}} \] Solving the given ode for \(y^{\prime }\) results in \(6\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\sqrt {x \left (\left (a^{2} x \right )^{\frac {1}{3}}-x \right )}}{x} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {x \left (\left (a^{2} x \right )^{\frac {1}{3}}-x \right )}}{x} \tag {2} \\ y^{\prime }&=\frac {\sqrt {2}\, \sqrt {x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-\left (a^{2} x \right )^{\frac {1}{3}}-2 x \right )}}{2 x} \tag {3} \\ y^{\prime }&=-\frac {\sqrt {2}\, \sqrt {x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-\left (a^{2} x \right )^{\frac {1}{3}}-2 x \right )}}{2 x} \tag {4} \\ y^{\prime }&=\frac {\sqrt {-2 x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}}{2 x} \tag {5} \\ y^{\prime }&=-\frac {\sqrt {-2 x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}}{2 x} \tag {6} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} y &= \int { \frac {\sqrt {x \left (\left (a^{2} x \right )^{\frac {1}{3}}-x \right )}}{x}\,\mathop {\mathrm {d}x}}\\ &= -\frac {\sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{4}}}\, \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{4}}}\, \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{4}}}\, \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \] Verified OK.
Solving equation (2)
Integrating both sides gives \begin {align*} y &= \int { -\frac {\sqrt {x \left (\left (a^{2} x \right )^{\frac {1}{3}}-x \right )}}{x}\,\mathop {\mathrm {d}x}}\\ &= \frac {\sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{4}}}\, \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{4}}}\, \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{4}}}\, \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{2} \] Verified OK.
Solving equation (3)
Integrating both sides gives \begin {align*} y &= \int { \frac {\sqrt {2}\, \sqrt {x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-\left (a^{2} x \right )^{\frac {1}{3}}-2 x \right )}}{2 x}\,\mathop {\mathrm {d}x}}\\ &= -\frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{3} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{3} \] Verified OK.
Solving equation (4)
Integrating both sides gives \begin {align*} y &= \int { -\frac {\sqrt {2}\, \sqrt {x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-\left (a^{2} x \right )^{\frac {1}{3}}-2 x \right )}}{2 x}\,\mathop {\mathrm {d}x}}\\ &= \frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{4} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{4} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{4} \] Verified OK.
Solving equation (5)
Integrating both sides gives \begin {align*} y &= \int { \frac {\sqrt {-2 x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}}{2 x}\,\mathop {\mathrm {d}x}}\\ &= \frac {\sqrt {-\frac {2 \left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{5} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {-\frac {2 \left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{5} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {-\frac {2 \left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{5} \] Verified OK.
Solving equation (6)
Integrating both sides gives \begin {align*} y &= \int { -\frac {\sqrt {-2 x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}}{2 x}\,\mathop {\mathrm {d}x}}\\ &= -\frac {\sqrt {-\frac {2 \left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{6} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\sqrt {-\frac {2 \left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{6} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\sqrt {-\frac {2 \left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{6} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left ({y^{\prime }}^{2}+1\right )^{3}=a^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {\sqrt {x \left (\left (a^{2} x \right )^{\frac {1}{3}}-x \right )}}{x}, y^{\prime }=-\frac {\sqrt {x \left (\left (a^{2} x \right )^{\frac {1}{3}}-x \right )}}{x}, y^{\prime }=-\frac {\sqrt {-2 x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}}{2 x}, y^{\prime }=\frac {\sqrt {-2 x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}}{2 x}, y^{\prime }=-\frac {\sqrt {2}\, \sqrt {x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-\left (a^{2} x \right )^{\frac {1}{3}}-2 x \right )}}{2 x}, y^{\prime }=\frac {\sqrt {2}\, \sqrt {x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-\left (a^{2} x \right )^{\frac {1}{3}}-2 x \right )}}{2 x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\sqrt {x \left (\left (a^{2} x \right )^{\frac {1}{3}}-x \right )}}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {\sqrt {x \left (\left (a^{2} x \right )^{\frac {1}{3}}-x \right )}}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (a^{2} x \right )^{\frac {2}{3}}-\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, a^{2}+c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{\left (a^{2} x \right )^{\frac {2}{3}}} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\sqrt {x \left (\left (a^{2} x \right )^{\frac {1}{3}}-x \right )}}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {\sqrt {x \left (\left (a^{2} x \right )^{\frac {1}{3}}-x \right )}}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (a^{2} x \right )^{\frac {2}{3}}-\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, a^{2}-c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{\left (a^{2} x \right )^{\frac {2}{3}}} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\sqrt {-2 x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}}{2 x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {\sqrt {-2 x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}}{2 x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {\sqrt {-2 x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\mathrm {I} \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (\mathrm {I} \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \sqrt {x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\sqrt {-2 x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}}{2 x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {\sqrt {-2 x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}}{2 x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {\sqrt {-2 x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\mathrm {I} \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (\mathrm {I} \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \sqrt {x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\sqrt {2}\, \sqrt {x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-\left (a^{2} x \right )^{\frac {1}{3}}-2 x \right )}}{2 x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {\sqrt {2}\, \sqrt {x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-\left (a^{2} x \right )^{\frac {1}{3}}-2 x \right )}}{2 x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\mathrm {I} \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (\mathrm {I} \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\sqrt {2}\, \sqrt {x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-\left (a^{2} x \right )^{\frac {1}{3}}-2 x \right )}}{2 x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {\sqrt {2}\, \sqrt {x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-\left (a^{2} x \right )^{\frac {1}{3}}-2 x \right )}}{2 x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\mathrm {I} \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (\mathrm {I} \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (a^{2} x \right )^{\frac {2}{3}}-\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, a^{2}+c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{\left (a^{2} x \right )^{\frac {2}{3}}}, y=-\frac {\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (a^{2} x \right )^{\frac {2}{3}}-\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, a^{2}-c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{\left (a^{2} x \right )^{\frac {2}{3}}}, y=-\frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\mathrm {I} \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (\mathrm {I} \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} , y=\frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\mathrm {I} \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (\mathrm {I} \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} , y=-\frac {\sqrt {-2 x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\mathrm {I} \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (\mathrm {I} \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \sqrt {x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} , y=\frac {\sqrt {-2 x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\mathrm {I} \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (\mathrm {I} \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \sqrt {x \left (\mathrm {I} \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \right \} \end {array} \]
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables <- differential order: 1; missing y(x) successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 601
dsolve(x^2*(diff(y(x),x)^2+1)^3-a^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {-\sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}\, a^{2}+c_{1} \left (a^{2} x \right )^{\frac {2}{3}}+\left (a^{2} x \right )^{\frac {2}{3}} \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}}{\left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right ) \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}+c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{\left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}-\sqrt {3}\, a^{2}\right )}{a^{2}}}\, \left (2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}+i \sqrt {3}\, a^{2}\right )}{4 \sqrt {\left (i \left (a^{2} x \right )^{\frac {1}{3}}+2 i x -\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}\right ) x}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}-\sqrt {3}\, a^{2}\right )}{a^{2}}}\, \left (2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}+i \sqrt {3}\, a^{2}\right )}{4 \sqrt {\left (i \left (a^{2} x \right )^{\frac {1}{3}}+2 i x -\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}\right ) x}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \left (x \right ) &= \frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) \sqrt {2}\, \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) x \left (a^{2} x \right )^{\frac {1}{3}}}{a^{2}}}+4 c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{4 \left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (-2 \left (a^{2} x \right )^{\frac {2}{3}} \sqrt {2}+a^{2} \left (i \sqrt {6}-\sqrt {2}\right )\right ) \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) x \left (a^{2} x \right )^{\frac {1}{3}}}{a^{2}}}-4 c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{4 \left (a^{2} x \right )^{\frac {2}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 19.706 (sec). Leaf size: 375
DSolve[-a^2 + x^2*(1 + y'[x]^2)^3==0,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*}