60.1.50 problem 51

Internal problem ID [10064]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 51
Date solved : Sunday, March 30, 2025 at 02:58:50 PM
CAS classification : [_Abel]

\begin{align*} y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )}&=0 \end{align*}

✓ Maple. Time used: 0.110 (sec). Leaf size: 648

ode:=diff(y(x),x)-(y(x)-f(x))*(y(x)-g(x))*(y(x)-(a*f(x)+b*g(x))/(a+b))*h(x)-diff(f(x),x)*(y(x)-g(x))/(f(x)-g(x))-diff(g(x),x)*(y(x)-f(x))/(g(x)-f(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 1.313 (sec). Leaf size: 243

ode=D[y[x],x] - (y[x]-f[x])*(y[x]-g[x])*(y[x]-(a*f[x]+b*g[x])/(a+b))*h[x]- D[ f[x],x]*(y[x]-g[x])/(f[x]-g[x]) - D[ g[x],x]*(y[x]-f[x])/(g[x]-f[x])==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {-2 a f(x) h(x)-b f(x) h(x)-a g(x) h(x)-2 b g(x) h(x)}{a+b}+3 h(x) y(x)}{\sqrt [3]{\frac {(f(x)-g(x))^3 \left (2 a^3 h(x)^3-2 b^3 h(x)^3-3 a b^2 h(x)^3+3 a^2 b h(x)^3\right )}{(a+b)^3}}}}\frac {1}{K[1]^3-\frac {3 \left (a^2+b a+b^2\right ) K[1]}{(a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}}+1}dK[1]=\int _1^x\frac {\left (\frac {(f(K[2])-g(K[2]))^3 \left (2 a^3 h(K[2])^3-2 b^3 h(K[2])^3-3 a b^2 h(K[2])^3+3 a^2 b h(K[2])^3\right )}{(a+b)^3}\right )^{2/3}}{9 h(K[2])}dK[2]+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
f = Function("f") 
g = Function("g") 
ode = Eq((f(x) - y(x))*(-g(x) + y(x))*(y(x) - (a*f(x) + b*g(x))/(a + b))*h(x) + Derivative(y(x), x) - (-g(x) + y(x))*Derivative(f(x), x)/(f(x) - g(x)) - (-f(x) + y(x))*Derivative(g(x), x)/(-f(x) + g(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out