60.7.155 problem 1746 (book 6.155)

Internal problem ID [11744]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1746 (book 6.155)
Date solved : Monday, January 27, 2025 at 11:33:29 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} 2 \left (y-a \right ) y^{\prime \prime }+{y^{\prime }}^{2}+1&=0 \end{align*}

Solution by Maple

Time used: 0.097 (sec). Leaf size: 123

dsolve(2*(y(x)-a)*diff(diff(y(x),x),x)+diff(y(x),x)^2+1=0,y(x), singsol=all)
 
\begin{align*} -\sqrt {-\left (-y+a \right ) \left (-y+c_{1} +a \right )}+\frac {\arctan \left (\frac {2 y-2 a -c_{1}}{2 \sqrt {-\left (-y+a \right ) \left (-y+c_{1} +a \right )}}\right ) c_{1}}{2}-x -c_{2} &= 0 \\ \sqrt {-\left (-y+a \right ) \left (-y+c_{1} +a \right )}-\frac {\arctan \left (\frac {2 y-2 a -c_{1}}{2 \sqrt {-\left (-y+a \right ) \left (-y+c_{1} +a \right )}}\right ) c_{1}}{2}-x -c_{2} &= 0 \\ \end{align*}

Solution by Mathematica

Time used: 1.127 (sec). Leaf size: 775

DSolve[1 + D[y[x],x]^2 + 2*(-a + y[x])*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )-\sqrt {2} e^{3 c_1} \sqrt {e^{-2 c_1} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )} \arcsin \left (\sqrt {2} e^{-c_1} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )-\sqrt {2} e^{3 c_1} \sqrt {e^{-2 c_1} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )} \arcsin \left (\sqrt {2} e^{-c_1} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 (-c_1)}\right )-\sqrt {2} e^{3 (-c_1)} \sqrt {e^{-2 (-c_1)} \left (2 \text {$\#$1}-2 a+e^{2 (-c_1)}\right )} \arcsin \left (\sqrt {2} e^{-(-c_1)} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 (-c_1)}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 (-c_1)}\right )-\sqrt {2} e^{3 (-c_1)} \sqrt {e^{-2 (-c_1)} \left (2 \text {$\#$1}-2 a+e^{2 (-c_1)}\right )} \arcsin \left (\sqrt {2} e^{-(-c_1)} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 (-c_1)}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )-\sqrt {2} e^{3 c_1} \sqrt {e^{-2 c_1} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )} \arcsin \left (\sqrt {2} e^{-c_1} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )-\sqrt {2} e^{3 c_1} \sqrt {e^{-2 c_1} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )} \arcsin \left (\sqrt {2} e^{-c_1} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}\&\right ][x+c_2] \\ \end{align*}