60.7.193 problem 1815 (book 6.224)

Internal problem ID [11743]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1815 (book 6.224)
Date solved : Wednesday, March 05, 2025 at 02:48:19 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 99

ode:=h(y(x))*diff(diff(y(x),x),x)+a*h(y(x))*diff(y(x),x)^2+j(y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \int _{}^{y}\frac {{\mathrm e}^{2 a \textit {\_b}}}{\sqrt {{\mathrm e}^{2 a \textit {\_b}} \left (-2 \left (\int \frac {{\mathrm e}^{2 a \textit {\_b}} j \left (\textit {\_b} \right )}{h \left (\textit {\_b} \right )}d \textit {\_b} \right )+c_{1} \right )}}d \textit {\_b} -x -c_{2} &= 0 \\ -\int _{}^{y}\frac {{\mathrm e}^{2 a \textit {\_b}}}{\sqrt {{\mathrm e}^{2 a \textit {\_b}} \left (-2 \left (\int \frac {{\mathrm e}^{2 a \textit {\_b}} j \left (\textit {\_b} \right )}{h \left (\textit {\_b} \right )}d \textit {\_b} \right )+c_{1} \right )}}d \textit {\_b} -x -c_{2} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.569 (sec). Leaf size: 362

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {2 \int _1^{K[2]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]-c_1}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {2 \int _1^{K[3]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]-c_1}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {e^{2 a K[1]} j(K[1])}{h(K[1])}dK[1]}}dK[3]\&\right ][x+c_2] \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
j = Function("j") 
h = Function("h") 
ode = Eq(a*h(y(x))*Derivative(y(x), x)**2 + h(y(x))*Derivative(y(x), (x, 2)) + j(y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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