Internal problem ID [3795]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 36
Problem number: 1088.
ODE order: 1.
ODE degree: 4.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{4}+f \relax (x ) \left (y-a \right )^{3} \left (y-b \right )^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.781 (sec). Leaf size: 262
dsolve(diff(y(x),x)^4+f(x)*(y(x)-a)^3*(y(x)-b)^3 = 0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (-f \left (\textit {\_a} \right ) \left (-y \relax (x )+b \right )^{3} \left (-y \relax (x )+a \right )^{3}\right )^{\frac {1}{4}}}{\left (\left (-y \relax (x )+a \right ) \left (-y \relax (x )+b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +c_{1} = 0 \\ \int _{}^{y \relax (x )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}\frac {i \left (-f \left (\textit {\_a} \right ) \left (-y \relax (x )+b \right )^{3} \left (-y \relax (x )+a \right )^{3}\right )^{\frac {1}{4}}}{\left (\left (-y \relax (x )+a \right ) \left (-y \relax (x )+b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +c_{1} = 0 \\ \int _{}^{y \relax (x )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}-\frac {i \left (-f \left (\textit {\_a} \right ) \left (-y \relax (x )+b \right )^{3} \left (-y \relax (x )+a \right )^{3}\right )^{\frac {1}{4}}}{\left (\left (-y \relax (x )+a \right ) \left (-y \relax (x )+b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +c_{1} = 0 \\ \int _{}^{y \relax (x )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}\frac {\left (-f \left (\textit {\_a} \right ) \left (-y \relax (x )+b \right )^{3} \left (-y \relax (x )+a \right )^{3}\right )^{\frac {1}{4}}}{\left (\left (-y \relax (x )+a \right ) \left (-y \relax (x )+b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 83.471 (sec). Leaf size: 385
DSolve[(y'[x])^4 +f[x] (y[x]-a)^3 (y[x]-b)^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \text {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\&\right ]\left [\int _1^x-\sqrt [4]{-1} \sqrt [4]{f(K[1])}dK[1]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \text {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\&\right ]\left [\int _1^x\sqrt [4]{-1} \sqrt [4]{f(K[2])}dK[2]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \text {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\&\right ]\left [\int _1^x-(-1)^{3/4} \sqrt [4]{f(K[3])}dK[3]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \text {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\&\right ]\left [\int _1^x(-1)^{3/4} \sqrt [4]{f(K[4])}dK[4]+c_1\right ] \\ y(x)\to a \\ y(x)\to b \\ \end{align*}