Internal problem ID [4304]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 36
Problem number: 1089.
ODE order: 1.
ODE degree: 4.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} \left (y-c \right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 92
dsolve(diff(y(x),x)^4+f(x)*(y(x)-a)^3*(y(x)-b)^3*(y(x)-c)^2 = 0,y(x), singsol=all)
\[ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -b \right )^{\frac {3}{4}} \left (\textit {\_a} -a \right )^{\frac {3}{4}} \sqrt {\textit {\_a} -c}}d \textit {\_a} +\int _{}^{x}-\frac {\left (-f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right )^{2} \left (b -y \left (x \right )\right )^{3} \left (-y \left (x \right )+a \right )^{3}\right )^{\frac {1}{4}}}{\left (y \left (x \right )-b \right )^{\frac {3}{4}} \left (y \left (x \right )-a \right )^{\frac {3}{4}} \sqrt {y \left (x \right )-c}}d \textit {\_a} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 23.526 (sec). Leaf size: 562
DSolve[(y'[x])^4 +f[x] (y[x]-a)^3 (y[x]-b)^3 (y[x]-c)^2 ==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} (a-c)}\&\right ]\left [\int _1^x-\sqrt [4]{-1} \sqrt [4]{f(K[1])}dK[1]+c_1\right ] y(x)\to \text {InverseFunction}\left [\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} (a-c)}\&\right ]\left [\int _1^x\sqrt [4]{-1} \sqrt [4]{f(K[2])}dK[2]+c_1\right ] y(x)\to \text {InverseFunction}\left [\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} (a-c)}\&\right ]\left [\int _1^x-(-1)^{3/4} \sqrt [4]{f(K[3])}dK[3]+c_1\right ] y(x)\to \text {InverseFunction}\left [\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} (a-c)}\&\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 y(x)\to c \end{align*}