Internal problem ID [7631]
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]
\[ \boxed {y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {f \left (x \right ) a +b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 2428
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: 1.232 (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]
\[ \text {Solve}\left [-\frac {1}{3} (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3} \text {RootSum}\left [\text {$\#$1}^3 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}-3 \text {$\#$1} a^2-3 \text {$\#$1} a b-3 \text {$\#$1} b^2+(a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}\&,\frac {\log \left (\frac {\frac {-2 a f(x) h(x)-a g(x) h(x)-b f(x) h(x)-2 b g(x) h(x)}{a+b}+3 h(x) y(x)}{\sqrt [3]{\frac {(f(x)-g(x))^3 \left (2 a^3 h(x)^3+3 a^2 b h(x)^3-3 a b^2 h(x)^3-2 b^3 h(x)^3\right )}{(a+b)^3}}}-\text {$\#$1}\right )}{-\text {$\#$1}^2 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}+a^2+a b+b^2}\&\right ]=\int _1^x\frac {\left (\frac {(f(K[1])-g(K[1]))^3 \left (2 a^3 h(K[1])^3-2 b^3 h(K[1])^3-3 a b^2 h(K[1])^3+3 a^2 b h(K[1])^3\right )}{(a+b)^3}\right )^{2/3}}{9 h(K[1])}dK[1]+c_1,y(x)\right ] \]