71.1.95 problem 122 (page 179)

Internal problem ID [19271]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 122 (page 179)
Date solved : Thursday, October 02, 2025 at 04:03:57 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}-{y^{\prime }}^{4}&=0 \end{align*}

✓ Maple. Time used: 0.081 (sec). Leaf size: 105

ode:=y(x)*diff(diff(y(x),x),x)-diff(y(x),x)^2-diff(y(x),x)^4 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ c_1 \ln \left (\frac {c_1 \left (\sqrt {c_1^{2}-y^{2}}+c_1 \right )}{y}\right )+c_1 \ln \left (2\right )-\sqrt {c_1^{2}-y^{2}}-c_2 -x &= 0 \\ -c_1 \ln \left (\frac {c_1 \left (\sqrt {c_1^{2}-y^{2}}+c_1 \right )}{y}\right )-c_1 \ln \left (2\right )+\sqrt {c_1^{2}-y^{2}}-c_2 -x &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.529 (sec). Leaf size: 361

ode=y[x]*D[y[x],{x,2}]-D[y[x],x]^2-D[y[x],x]^4==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [-i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-i e^{-(-c_1)} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}\right )\right )\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [i e^{-(-c_1)} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}\right )\right )\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**4 - Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(sqrt(4*y(x)*Derivative(y(x), (x, 2)) + 1)/2 - 1/2) + Derivative(y(x), x) cannot be solved by the factorable group method