Internal problem ID [4246]
Internal file name [OUTPUT/3739_Sunday_June_05_2022_10_35_08_AM_28740706/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1021.
ODE order: 1.
ODE degree: 3.
The type(s) of ODE detected by this program : "first_order_nonlinear_p_but_separable"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {{y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2}=0} \]
The ode has the form \begin {align*} (y')^{\frac {n}{m}} &= f(x) g(y)\tag {1} \end {align*}
Where \(n=3, m=1, f=-f \left (x \right ) , g=\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\). Hence the ode is \begin {align*} (y')^{3} &= -f \left (x \right ) \left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2} \end {align*}
Solving for \(y^{\prime }\) from (1) gives \begin {align*} y^{\prime } &=\left (f g \right )^{\frac {1}{3}}\\ y^{\prime } &=-\frac {\left (f g \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (f g \right )^{\frac {1}{3}}}{2}\\ y^{\prime } &=-\frac {\left (f g \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (f g \right )^{\frac {1}{3}}}{2} \end {align*}
To be able to solve as separable ode, we have to now assume that \(f>0,g>0\). \begin {align*} -f \left (x \right ) &> 0\\ \left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2} &> 0 \end {align*}
Under the above assumption the differential equations become separable and can be written as \begin {align*} y^{\prime } &=f^{\frac {1}{3}} g^{\frac {1}{3}}\\ y^{\prime } &=\frac {f^{\frac {1}{3}} g^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2}\\ y^{\prime } &=-\frac {f^{\frac {1}{3}} g^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2} \end {align*}
Therefore \begin {align*} \frac {1}{g^{\frac {1}{3}}} \, dy &= \left (f^{\frac {1}{3}}\right )\,dx\\ \frac {2}{g^{\frac {1}{3}} \left (i \sqrt {3}-1\right )} \, dy &= \left (f^{\frac {1}{3}}\right )\,dx\\ -\frac {2}{g^{\frac {1}{3}} \left (1+i \sqrt {3}\right )} \, dy &= \left (f^{\frac {1}{3}}\right )\,dx \end {align*}
Replacing \(f(x),g(y)\) by their values gives \begin {align*} \frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}}} \, dy &= \left (\left (-f \left (x \right )\right )^{\frac {1}{3}}\right )\,dx\\ \frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )} \, dy &= \left (\left (-f \left (x \right )\right )^{\frac {1}{3}}\right )\,dx\\ -\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )} \, dy &= \left (\left (-f \left (x \right )\right )^{\frac {1}{3}}\right )\,dx \end {align*}
Integrating now gives the solutions. \begin {align*} \int \frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}}}d y &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}\\ \int \frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}d y &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}\\ \int -\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d y &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1} \end {align*}
Integrating gives \begin {align*} \int _{}^{y}\frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}\\ \int _{}^{y}\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}d \textit {\_a} &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}\\ \int _{}^{y}-\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d \textit {\_a} &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1} \end {align*}
Therefore \begin{align*} \int _{}^{y}\frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1} \\ \int _{}^{y}\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}d \textit {\_a} &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1} \\ \int _{}^{y}-\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d \textit {\_a} &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1} \\ \end{align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1} \\ \tag{2} \int _{}^{y}\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}d \textit {\_a} &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1} \\ \tag{3} \int _{}^{y}-\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d \textit {\_a} &= \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1} \\ \end{align*}
Verification of solutions
\[
\int _{}^{y}\frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK. {0 < (c-y)^2*(b-y)^2*(a-y)^2, 0 < -f(x)}
\[
\int _{}^{y}\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK. {0 < (c-y)^2*(b-y)^2*(a-y)^2, 0 < -f(x)}
\[
\int _{}^{y}-\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK. {0 < (c-y)^2*(b-y)^2*(a-y)^2, 0 < -f(x)}
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\left (2 f \left (x \right ) y^{5} b +2 f \left (x \right ) y^{5} a +2 f \left (x \right ) y^{5} c -f \left (x \right ) y^{4} a^{2}-f \left (x \right ) y^{4} b^{2}-f \left (x \right ) y^{4} c^{2}-4 f \left (x \right ) y^{4} a b -4 f \left (x \right ) y^{4} a c -4 f \left (x \right ) y^{4} b c +2 f \left (x \right ) y^{3} a^{2} b +2 f \left (x \right ) y^{3} a^{2} c +2 f \left (x \right ) y^{3} a \,b^{2}+2 f \left (x \right ) y^{3} a \,c^{2}+2 f \left (x \right ) y^{3} b^{2} c +2 f \left (x \right ) y^{3} b \,c^{2}-f \left (x \right ) y^{2} a^{2} b^{2}-f \left (x \right ) y^{2} a^{2} c^{2}-f \left (x \right ) y^{2} b^{2} c^{2}-f \left (x \right ) a^{2} b^{2} c^{2}+2 f \left (x \right ) y a^{2} b^{2} c +2 f \left (x \right ) y a^{2} b \,c^{2}+2 f \left (x \right ) y a \,b^{2} c^{2}-4 f \left (x \right ) y^{2} a^{2} b c -4 f \left (x \right ) y^{2} a \,b^{2} c -4 f \left (x \right ) y^{2} a b \,c^{2}+8 f \left (x \right ) y^{3} a b c -f \left (x \right ) y^{6}\right )^{\frac {1}{3}}, y^{\prime }=-\frac {\left (2 f \left (x \right ) y^{5} b +2 f \left (x \right ) y^{5} a +2 f \left (x \right ) y^{5} c -f \left (x \right ) y^{4} a^{2}-f \left (x \right ) y^{4} b^{2}-f \left (x \right ) y^{4} c^{2}-4 f \left (x \right ) y^{4} a b -4 f \left (x \right ) y^{4} a c -4 f \left (x \right ) y^{4} b c +2 f \left (x \right ) y^{3} a^{2} b +2 f \left (x \right ) y^{3} a^{2} c +2 f \left (x \right ) y^{3} a \,b^{2}+2 f \left (x \right ) y^{3} a \,c^{2}+2 f \left (x \right ) y^{3} b^{2} c +2 f \left (x \right ) y^{3} b \,c^{2}-f \left (x \right ) y^{2} a^{2} b^{2}-f \left (x \right ) y^{2} a^{2} c^{2}-f \left (x \right ) y^{2} b^{2} c^{2}-f \left (x \right ) a^{2} b^{2} c^{2}+2 f \left (x \right ) y a^{2} b^{2} c +2 f \left (x \right ) y a^{2} b \,c^{2}+2 f \left (x \right ) y a \,b^{2} c^{2}-4 f \left (x \right ) y^{2} a^{2} b c -4 f \left (x \right ) y^{2} a \,b^{2} c -4 f \left (x \right ) y^{2} a b \,c^{2}+8 f \left (x \right ) y^{3} a b c -f \left (x \right ) y^{6}\right )^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, \left (2 f \left (x \right ) y^{5} b +2 f \left (x \right ) y^{5} a +2 f \left (x \right ) y^{5} c -f \left (x \right ) y^{4} a^{2}-f \left (x \right ) y^{4} b^{2}-f \left (x \right ) y^{4} c^{2}-4 f \left (x \right ) y^{4} a b -4 f \left (x \right ) y^{4} a c -4 f \left (x \right ) y^{4} b c +2 f \left (x \right ) y^{3} a^{2} b +2 f \left (x \right ) y^{3} a^{2} c +2 f \left (x \right ) y^{3} a \,b^{2}+2 f \left (x \right ) y^{3} a \,c^{2}+2 f \left (x \right ) y^{3} b^{2} c +2 f \left (x \right ) y^{3} b \,c^{2}-f \left (x \right ) y^{2} a^{2} b^{2}-f \left (x \right ) y^{2} a^{2} c^{2}-f \left (x \right ) y^{2} b^{2} c^{2}-f \left (x \right ) a^{2} b^{2} c^{2}+2 f \left (x \right ) y a^{2} b^{2} c +2 f \left (x \right ) y a^{2} b \,c^{2}+2 f \left (x \right ) y a \,b^{2} c^{2}-4 f \left (x \right ) y^{2} a^{2} b c -4 f \left (x \right ) y^{2} a \,b^{2} c -4 f \left (x \right ) y^{2} a b \,c^{2}+8 f \left (x \right ) y^{3} a b c -f \left (x \right ) y^{6}\right )^{\frac {1}{3}}}{2}, y^{\prime }=-\frac {\left (2 f \left (x \right ) y^{5} b +2 f \left (x \right ) y^{5} a +2 f \left (x \right ) y^{5} c -f \left (x \right ) y^{4} a^{2}-f \left (x \right ) y^{4} b^{2}-f \left (x \right ) y^{4} c^{2}-4 f \left (x \right ) y^{4} a b -4 f \left (x \right ) y^{4} a c -4 f \left (x \right ) y^{4} b c +2 f \left (x \right ) y^{3} a^{2} b +2 f \left (x \right ) y^{3} a^{2} c +2 f \left (x \right ) y^{3} a \,b^{2}+2 f \left (x \right ) y^{3} a \,c^{2}+2 f \left (x \right ) y^{3} b^{2} c +2 f \left (x \right ) y^{3} b \,c^{2}-f \left (x \right ) y^{2} a^{2} b^{2}-f \left (x \right ) y^{2} a^{2} c^{2}-f \left (x \right ) y^{2} b^{2} c^{2}-f \left (x \right ) a^{2} b^{2} c^{2}+2 f \left (x \right ) y a^{2} b^{2} c +2 f \left (x \right ) y a^{2} b \,c^{2}+2 f \left (x \right ) y a \,b^{2} c^{2}-4 f \left (x \right ) y^{2} a^{2} b c -4 f \left (x \right ) y^{2} a \,b^{2} c -4 f \left (x \right ) y^{2} a b \,c^{2}+8 f \left (x \right ) y^{3} a b c -f \left (x \right ) y^{6}\right )^{\frac {1}{3}}}{2}+\frac {\mathrm {I} \sqrt {3}\, \left (2 f \left (x \right ) y^{5} b +2 f \left (x \right ) y^{5} a +2 f \left (x \right ) y^{5} c -f \left (x \right ) y^{4} a^{2}-f \left (x \right ) y^{4} b^{2}-f \left (x \right ) y^{4} c^{2}-4 f \left (x \right ) y^{4} a b -4 f \left (x \right ) y^{4} a c -4 f \left (x \right ) y^{4} b c +2 f \left (x \right ) y^{3} a^{2} b +2 f \left (x \right ) y^{3} a^{2} c +2 f \left (x \right ) y^{3} a \,b^{2}+2 f \left (x \right ) y^{3} a \,c^{2}+2 f \left (x \right ) y^{3} b^{2} c +2 f \left (x \right ) y^{3} b \,c^{2}-f \left (x \right ) y^{2} a^{2} b^{2}-f \left (x \right ) y^{2} a^{2} c^{2}-f \left (x \right ) y^{2} b^{2} c^{2}-f \left (x \right ) a^{2} b^{2} c^{2}+2 f \left (x \right ) y a^{2} b^{2} c +2 f \left (x \right ) y a^{2} b \,c^{2}+2 f \left (x \right ) y a \,b^{2} c^{2}-4 f \left (x \right ) y^{2} a^{2} b c -4 f \left (x \right ) y^{2} a \,b^{2} c -4 f \left (x \right ) y^{2} a b \,c^{2}+8 f \left (x \right ) y^{3} a b c -f \left (x \right ) y^{6}\right )^{\frac {1}{3}}}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\left (2 f \left (x \right ) y^{5} b +2 f \left (x \right ) y^{5} a +2 f \left (x \right ) y^{5} c -f \left (x \right ) y^{4} a^{2}-f \left (x \right ) y^{4} b^{2}-f \left (x \right ) y^{4} c^{2}-4 f \left (x \right ) y^{4} a b -4 f \left (x \right ) y^{4} a c -4 f \left (x \right ) y^{4} b c +2 f \left (x \right ) y^{3} a^{2} b +2 f \left (x \right ) y^{3} a^{2} c +2 f \left (x \right ) y^{3} a \,b^{2}+2 f \left (x \right ) y^{3} a \,c^{2}+2 f \left (x \right ) y^{3} b^{2} c +2 f \left (x \right ) y^{3} b \,c^{2}-f \left (x \right ) y^{2} a^{2} b^{2}-f \left (x \right ) y^{2} a^{2} c^{2}-f \left (x \right ) y^{2} b^{2} c^{2}-f \left (x \right ) a^{2} b^{2} c^{2}+2 f \left (x \right ) y a^{2} b^{2} c +2 f \left (x \right ) y a^{2} b \,c^{2}+2 f \left (x \right ) y a \,b^{2} c^{2}-4 f \left (x \right ) y^{2} a^{2} b c -4 f \left (x \right ) y^{2} a \,b^{2} c -4 f \left (x \right ) y^{2} a b \,c^{2}+8 f \left (x \right ) y^{3} a b c -f \left (x \right ) y^{6}\right )^{\frac {1}{3}} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (2 f \left (x \right ) y^{5} b +2 f \left (x \right ) y^{5} a +2 f \left (x \right ) y^{5} c -f \left (x \right ) y^{4} a^{2}-f \left (x \right ) y^{4} b^{2}-f \left (x \right ) y^{4} c^{2}-4 f \left (x \right ) y^{4} a b -4 f \left (x \right ) y^{4} a c -4 f \left (x \right ) y^{4} b c +2 f \left (x \right ) y^{3} a^{2} b +2 f \left (x \right ) y^{3} a^{2} c +2 f \left (x \right ) y^{3} a \,b^{2}+2 f \left (x \right ) y^{3} a \,c^{2}+2 f \left (x \right ) y^{3} b^{2} c +2 f \left (x \right ) y^{3} b \,c^{2}-f \left (x \right ) y^{2} a^{2} b^{2}-f \left (x \right ) y^{2} a^{2} c^{2}-f \left (x \right ) y^{2} b^{2} c^{2}-f \left (x \right ) a^{2} b^{2} c^{2}+2 f \left (x \right ) y a^{2} b^{2} c +2 f \left (x \right ) y a^{2} b \,c^{2}+2 f \left (x \right ) y a \,b^{2} c^{2}-4 f \left (x \right ) y^{2} a^{2} b c -4 f \left (x \right ) y^{2} a \,b^{2} c -4 f \left (x \right ) y^{2} a b \,c^{2}+8 f \left (x \right ) y^{3} a b c -f \left (x \right ) y^{6}\right )^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \sqrt {3}\, \left (2 f \left (x \right ) y^{5} b +2 f \left (x \right ) y^{5} a +2 f \left (x \right ) y^{5} c -f \left (x \right ) y^{4} a^{2}-f \left (x \right ) y^{4} b^{2}-f \left (x \right ) y^{4} c^{2}-4 f \left (x \right ) y^{4} a b -4 f \left (x \right ) y^{4} a c -4 f \left (x \right ) y^{4} b c +2 f \left (x \right ) y^{3} a^{2} b +2 f \left (x \right ) y^{3} a^{2} c +2 f \left (x \right ) y^{3} a \,b^{2}+2 f \left (x \right ) y^{3} a \,c^{2}+2 f \left (x \right ) y^{3} b^{2} c +2 f \left (x \right ) y^{3} b \,c^{2}-f \left (x \right ) y^{2} a^{2} b^{2}-f \left (x \right ) y^{2} a^{2} c^{2}-f \left (x \right ) y^{2} b^{2} c^{2}-f \left (x \right ) a^{2} b^{2} c^{2}+2 f \left (x \right ) y a^{2} b^{2} c +2 f \left (x \right ) y a^{2} b \,c^{2}+2 f \left (x \right ) y a \,b^{2} c^{2}-4 f \left (x \right ) y^{2} a^{2} b c -4 f \left (x \right ) y^{2} a \,b^{2} c -4 f \left (x \right ) y^{2} a b \,c^{2}+8 f \left (x \right ) y^{3} a b c -f \left (x \right ) y^{6}\right )^{\frac {1}{3}}}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (2 f \left (x \right ) y^{5} b +2 f \left (x \right ) y^{5} a +2 f \left (x \right ) y^{5} c -f \left (x \right ) y^{4} a^{2}-f \left (x \right ) y^{4} b^{2}-f \left (x \right ) y^{4} c^{2}-4 f \left (x \right ) y^{4} a b -4 f \left (x \right ) y^{4} a c -4 f \left (x \right ) y^{4} b c +2 f \left (x \right ) y^{3} a^{2} b +2 f \left (x \right ) y^{3} a^{2} c +2 f \left (x \right ) y^{3} a \,b^{2}+2 f \left (x \right ) y^{3} a \,c^{2}+2 f \left (x \right ) y^{3} b^{2} c +2 f \left (x \right ) y^{3} b \,c^{2}-f \left (x \right ) y^{2} a^{2} b^{2}-f \left (x \right ) y^{2} a^{2} c^{2}-f \left (x \right ) y^{2} b^{2} c^{2}-f \left (x \right ) a^{2} b^{2} c^{2}+2 f \left (x \right ) y a^{2} b^{2} c +2 f \left (x \right ) y a^{2} b \,c^{2}+2 f \left (x \right ) y a \,b^{2} c^{2}-4 f \left (x \right ) y^{2} a^{2} b c -4 f \left (x \right ) y^{2} a \,b^{2} c -4 f \left (x \right ) y^{2} a b \,c^{2}+8 f \left (x \right ) y^{3} a b c -f \left (x \right ) y^{6}\right )^{\frac {1}{3}}}{2}+\frac {\mathrm {I} \sqrt {3}\, \left (2 f \left (x \right ) y^{5} b +2 f \left (x \right ) y^{5} a +2 f \left (x \right ) y^{5} c -f \left (x \right ) y^{4} a^{2}-f \left (x \right ) y^{4} b^{2}-f \left (x \right ) y^{4} c^{2}-4 f \left (x \right ) y^{4} a b -4 f \left (x \right ) y^{4} a c -4 f \left (x \right ) y^{4} b c +2 f \left (x \right ) y^{3} a^{2} b +2 f \left (x \right ) y^{3} a^{2} c +2 f \left (x \right ) y^{3} a \,b^{2}+2 f \left (x \right ) y^{3} a \,c^{2}+2 f \left (x \right ) y^{3} b^{2} c +2 f \left (x \right ) y^{3} b \,c^{2}-f \left (x \right ) y^{2} a^{2} b^{2}-f \left (x \right ) y^{2} a^{2} c^{2}-f \left (x \right ) y^{2} b^{2} c^{2}-f \left (x \right ) a^{2} b^{2} c^{2}+2 f \left (x \right ) y a^{2} b^{2} c +2 f \left (x \right ) y a^{2} b \,c^{2}+2 f \left (x \right ) y a \,b^{2} c^{2}-4 f \left (x \right ) y^{2} a^{2} b c -4 f \left (x \right ) y^{2} a \,b^{2} c -4 f \left (x \right ) y^{2} a b \,c^{2}+8 f \left (x \right ) y^{3} a b c -f \left (x \right ) y^{6}\right )^{\frac {1}{3}}}{2} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying simple symmetries for implicit equations Successful isolation of dy/dx: 3 solutions were found. Trying to solve each resulting ODE. *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying exact <- exact successful ------------------- * Tackling next ODE. *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying exact <- exact successful ------------------- * Tackling next ODE. *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying exact <- exact successful`
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 269
dsolve(diff(y(x),x)^3+f(x)*(y(x)-a)^2*(y(x)-b)^2*(y(x)-c)^2 = 0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -c \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {2}{3}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right )^{2} \left (y \left (x \right )-b \right )^{2} \left (y \left (x \right )-a \right )^{2}\right )^{\frac {1}{3}}d \textit {\_a}}{\left (\left (y \left (x \right )-c \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -c \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {2}{3}}}d \textit {\_a} +\frac {\left (1+i \sqrt {3}\right ) \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right )^{2} \left (y \left (x \right )-b \right )^{2} \left (y \left (x \right )-a \right )^{2}\right )^{\frac {1}{3}}d \textit {\_a} \right )}{2 \left (\left (y \left (x \right )-c \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -c \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {2}{3}}}d \textit {\_a} -\frac {\left (-1+i \sqrt {3}\right ) \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right )^{2} \left (y \left (x \right )-b \right )^{2} \left (y \left (x \right )-a \right )^{2}\right )^{\frac {1}{3}}d \textit {\_a} \right )}{2 \left (\left (y \left (x \right )-c \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 15.995 (sec). Leaf size: 421
DSolve[(y'[x])^3 +f[x](y[x]-a)^2 (y[x]-b)^2 (y[x]-c)^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt [3]{c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (a-c)}\&\right ]\left [\int _1^x-\sqrt [3]{f(K[1])}dK[1]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt [3]{c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (a-c)}\&\right ]\left [\int _1^x\sqrt [3]{-1} \sqrt [3]{f(K[2])}dK[2]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt [3]{c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (a-c)}\&\right ]\left [\int _1^x-(-1)^{2/3} \sqrt [3]{f(K[3])}dK[3]+c_1\right ] \\ y(x)\to a \\ y(x)\to b \\ y(x)\to c \\ \end{align*}