Internal problem ID [4170]
Book: Differential Gleichungen, Kamke, 3rd ed, Abel ODEs
Section: Abel ODE’s with constant invariant
Problem number: problem 51.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Abel]
Solve \begin {gather*} \boxed {y^{\prime }-\left (y-f \relax (x )\right ) \left (y-g \relax (x )\right ) \left (y-\frac {a f \relax (x )+b g \relax (x )}{a +b}\right ) h \relax (x )-\frac {f^{\prime }\relax (x ) \left (y-g \relax (x )\right )}{f \relax (x )-g \relax (x )}-\frac {g^{\prime }\relax (x ) \left (y-f \relax (x )\right )}{g \relax (x )-f \relax (x )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.127 (sec). Leaf size: 1135
dsolve(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,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 1.4 (sec). Leaf size: 355
DSolve[y'[x]-(y[x]-f[x])*(y[x]-g[x])*(y[x]-(a*f[x]+b*g[x])/(a+b))*h[x]-f'[x]*(y[x]-g[x])/(f[x]-g[x])-g'[x]*(y[x]-f[x])/(g[x]-f[x]) == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {1}{3} (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3} \text {RootSum}\left [\text {$\#$1}^3 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}-3 \text {$\#$1} a^2-3 \text {$\#$1} a b-3 \text {$\#$1} b^2+(a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}\&,\frac {\log \left (\frac {\frac {-2 a f(x) h(x)-a g(x) h(x)-b f(x) h(x)-2 b g(x) h(x)}{a+b}+3 h(x) y(x)}{\sqrt [3]{\frac {(f(x)-g(x))^3 \left (2 a^3 h(x)^3+3 a^2 b h(x)^3-3 a b^2 h(x)^3-2 b^3 h(x)^3\right )}{(a+b)^3}}}-\text {$\#$1}\right )}{-\text {$\#$1}^2 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}+a^2+a b+b^2}\&\right ]=\int _1^x\frac {\left (\frac {(f(K[1])-g(K[1]))^3 \left (2 a^3 h(K[1])^3-2 b^3 h(K[1])^3-3 a b^2 h(K[1])^3+3 a^2 b h(K[1])^3\right )}{(a+b)^3}\right )^{2/3}}{9 h(K[1])}dK[1]+c_1,y(x)\right ] \]