Internal problem ID [4245]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1020.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {{y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 212
dsolve(diff(y(x),x)^3+f(x)*(y(x)-a)^2*(y(x)-b)^2 = 0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {2}{3}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (-f \left (\textit {\_a} \right ) \left (-y \left (x \right )+a \right )^{2} \left (b -y \left (x \right )\right )^{2}\right )^{\frac {1}{3}}}{\left (\left (-y \left (x \right )+a \right ) \left (b -y \left (x \right )\right )\right )^{\frac {2}{3}}}d \textit {\_a} +c_{1} = 0 \int _{}^{y \left (x \right )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {2}{3}}}d \textit {\_a} +\int _{}^{x}\frac {\left (-f \left (\textit {\_a} \right ) \left (-y \left (x \right )+a \right )^{2} \left (b -y \left (x \right )\right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2 \left (\left (-y \left (x \right )+a \right ) \left (b -y \left (x \right )\right )\right )^{\frac {2}{3}}}d \textit {\_a} +c_{1} = 0 \int _{}^{y \left (x \right )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {2}{3}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (-f \left (\textit {\_a} \right ) \left (-y \left (x \right )+a \right )^{2} \left (b -y \left (x \right )\right )^{2}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{2 \left (\left (-y \left (x \right )+a \right ) \left (b -y \left (x \right )\right )\right )^{\frac {2}{3}}}d \textit {\_a} +c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 1.083 (sec). Leaf size: 287
DSolve[(y'[x])^3 +f[x] (y[x]-a)^2 (y[x]-b)^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-\sqrt [3]{f(K[1])}dK[1]+c_1\right ] y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x\sqrt [3]{-1} \sqrt [3]{f(K[2])}dK[2]+c_1\right ] y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\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 \end{align*}