Internal problem ID [4302]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 36
Problem number: 1087.
ODE order: 1.
ODE degree: 4.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 270
dsolve(diff(y(x),x)^4+f(x)*(y(x)-a)^3*(y(x)-b)^2 = 0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} -b}\, \left (\textit {\_a} -a \right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (f \left (\textit {\_a} \right ) \left (-y \left (x \right )+a \right )^{3} \left (b -y \left (x \right )\right )^{2}\right )^{\frac {1}{4}}}{\sqrt {y \left (x \right )-b}\, \left (y \left (x \right )-a \right )^{\frac {3}{4}}}d \textit {\_a} +c_{1} = 0 \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} -b}\, \left (\textit {\_a} -a \right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}\frac {i \left (f \left (\textit {\_a} \right ) \left (-y \left (x \right )+a \right )^{3} \left (b -y \left (x \right )\right )^{2}\right )^{\frac {1}{4}}}{\sqrt {y \left (x \right )-b}\, \left (y \left (x \right )-a \right )^{\frac {3}{4}}}d \textit {\_a} +c_{1} = 0 \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} -b}\, \left (\textit {\_a} -a \right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}-\frac {i \left (f \left (\textit {\_a} \right ) \left (-y \left (x \right )+a \right )^{3} \left (b -y \left (x \right )\right )^{2}\right )^{\frac {1}{4}}}{\sqrt {y \left (x \right )-b}\, \left (y \left (x \right )-a \right )^{\frac {3}{4}}}d \textit {\_a} +c_{1} = 0 \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} -b}\, \left (\textit {\_a} -a \right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}\frac {\left (f \left (\textit {\_a} \right ) \left (-y \left (x \right )+a \right )^{3} \left (b -y \left (x \right )\right )^{2}\right )^{\frac {1}{4}}}{\sqrt {y \left (x \right )-b}\, \left (y \left (x \right )-a \right )^{\frac {3}{4}}}d \textit {\_a} +c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 1.878 (sec). Leaf size: 369
DSolve[(y'[x])^4 +f[x] (y[x]-a)^3 (y[x]-b)^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x-\sqrt [4]{f(K[1])}dK[1]+c_1\right ] y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x-i \sqrt [4]{f(K[2])}dK[2]+c_1\right ] y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^xi \sqrt [4]{f(K[3])}dK[3]+c_1\right ] y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x\sqrt [4]{f(K[4])}dK[4]+c_1\right ] y(x)\to a y(x)\to b \end{align*}