Internal problem ID [8388]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 51.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Abel]
\[ \boxed {y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 649
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)
\[ y \left (x \right ) = \frac {2 \left (f \left (x \right )-g \left (x \right )\right ) \left (a +\frac {b}{2}\right ) {\mathrm e}^{\operatorname {RootOf}\left (-2 a^{3} b \left (\int g \left (x \right ) f \left (x \right ) h \left (x \right )d x \right )-2 a^{2} b^{2} \left (\int g \left (x \right ) f \left (x \right ) h \left (x \right )d x \right )-2 a \,b^{3} \left (\int g \left (x \right ) f \left (x \right ) h \left (x \right )d x \right )+a^{3} b \left (\int f \left (x \right )^{2} h \left (x \right )d x \right )+a^{2} b^{2} \left (\int f \left (x \right )^{2} h \left (x \right )d x \right )+a \,b^{3} \left (\int f \left (x \right )^{2} h \left (x \right )d x \right )+a^{3} b \left (\int g \left (x \right )^{2} h \left (x \right )d x \right )+a^{2} b^{2} \left (\int g \left (x \right )^{2} h \left (x \right )d x \right )+a \,b^{3} \left (\int g \left (x \right )^{2} h \left (x \right )d x \right )-2 a^{3} b \ln \left (\frac {-9 a^{3}-18 a^{2} b -18 a \,b^{2}-9 b^{3}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a +2 b}\right )-2 a^{2} b^{2} \ln \left (\frac {-9 a^{3}-18 a^{2} b -18 a \,b^{2}-9 b^{3}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a +2 b}\right )-a \,b^{3} \ln \left (\frac {-9 a^{3}-18 a^{2} b -18 a \,b^{2}-9 b^{3}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a +2 b}\right )+3 \ln \left (\frac {-9 a^{3}-9 a^{2} b -9 a \,b^{2}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{3} b +4 \ln \left (\frac {-9 a^{3}-9 a^{2} b -9 a \,b^{2}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{2} b^{2}+3 \ln \left (\frac {-9 a^{3}-9 a^{2} b -9 a \,b^{2}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a \,b^{3}-\textit {\_Z} \,a^{3} b -2 \textit {\_Z} \,a^{2} b^{2}-2 \textit {\_Z} a \,b^{3}+3 c_{1} a^{3} b +6 c_{1} a^{2} b^{2}+3 c_{1} a \,b^{3}-a^{4} \ln \left (\frac {-9 a^{3}-18 a^{2} b -18 a \,b^{2}-9 b^{3}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a +2 b}\right )+\ln \left (\frac {-9 a^{3}-9 a^{2} b -9 a \,b^{2}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{4}+\ln \left (\frac {-9 a^{3}-9 a^{2} b -9 a \,b^{2}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) b^{4}-\textit {\_Z} \,b^{4}\right )}+9 \left (a +b \right ) \left (a^{2}+a b +b^{2}\right ) g \left (x \right )}{9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}} \]
✓ Solution by Mathematica
Time used: 1.168 (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]
\[ \text {Solve}\left [-\frac {1}{3} (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3} \text {RootSum}\left [\text {$\#$1}^3 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}-3 \text {$\#$1} a^2-3 \text {$\#$1} a b-3 \text {$\#$1} b^2+(a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}\&,\frac {\log \left (\frac {\frac {-2 a f(x) h(x)-a g(x) h(x)-b f(x) h(x)-2 b g(x) h(x)}{a+b}+3 h(x) y(x)}{\sqrt [3]{\frac {(f(x)-g(x))^3 \left (2 a^3 h(x)^3+3 a^2 b h(x)^3-3 a b^2 h(x)^3-2 b^3 h(x)^3\right )}{(a+b)^3}}}-\text {$\#$1}\right )}{-\text {$\#$1}^2 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}+a^2+a b+b^2}\&\right ]=\int _1^x\frac {\left (\frac {(f(K[1])-g(K[1]))^3 \left (2 a^3 h(K[1])^3-2 b^3 h(K[1])^3-3 a b^2 h(K[1])^3+3 a^2 b h(K[1])^3\right )}{(a+b)^3}\right )^{2/3}}{9 h(K[1])}dK[1]+c_1,y(x)\right ] \]