60.1.51 problem 51

Internal problem ID [10065]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 51
Date solved : Tuesday, January 28, 2025 at 04:15:38 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*}

✓ Solution by Maple

Time used: 1.646 (sec). Leaf size: 648

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 = \frac {\left (f-g \left (x \right )\right ) \left (a +2 b \right ) {\mathrm e}^{\operatorname {RootOf}\left (\ln \left (\frac {9 a^{2} b +9 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{4}+\ln \left (\frac {9 a^{2} b +9 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) b^{4}-b^{4} \ln \left (\frac {9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{2 a +b}\right )-\textit {\_Z} \,a^{4}+a^{2} b^{2} \left (\int h \left (x \right ) g \left (x \right )^{2}d x \right )+a \,b^{3} \left (\int h \left (x \right ) g \left (x \right )^{2}d x \right )+3 c_{1} a^{3} b +6 c_{1} a^{2} b^{2}+3 c_{1} a \,b^{3}+a^{3} b \left (\int h \left (x \right ) f^{2}d x \right )+a^{2} b^{2} \left (\int h \left (x \right ) f^{2}d x \right )+a \,b^{3} \left (\int h \left (x \right ) f^{2}d x \right )+a^{3} b \left (\int h \left (x \right ) g \left (x \right )^{2}d x \right )-2 a^{3} b \left (\int h \left (x \right ) g \left (x \right ) fd x \right )-2 a^{2} b^{2} \left (\int h \left (x \right ) g \left (x \right ) fd x \right )-2 a \,b^{3} \left (\int h \left (x \right ) g \left (x \right ) fd x \right )+3 \ln \left (\frac {9 a^{2} b +9 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{3} b +4 \ln \left (\frac {9 a^{2} b +9 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{2} b^{2}+3 \ln \left (\frac {9 a^{2} b +9 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a \,b^{3}-a^{3} b \ln \left (\frac {9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{2 a +b}\right )-2 a^{2} b^{2} \ln \left (\frac {9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{2 a +b}\right )-2 a \,b^{3} \ln \left (\frac {9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{2 a +b}\right )-2 \textit {\_Z} \,a^{3} b -2 \textit {\_Z} \,a^{2} b^{2}-\textit {\_Z} a \,b^{3}\right )}+9 \left (a +b \right ) \left (a^{2}+a b +b^{2}\right ) f}{9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}} \]

✓ Solution by Mathematica

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

DSolve[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,y[x],x,IncludeSingularSolutions -> True]
 

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