29.36.19 problem 1087
Internal
problem
ID
[5646]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
36
Problem
number
:
1087
Date
solved
:
Sunday, March 30, 2025 at 09:52:50 AM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} {y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{2}&=0 \end{align*}
✓ Maple. Time used: 0.156 (sec). Leaf size: 272
ode:=diff(y(x),x)^4+f(x)*(y(x)-a)^3*(y(x)-b)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\int _{}^{y}\frac {1}{\left (\textit {\_a} -a \right )^{{3}/{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y-b \right )^{2} \left (y-a \right )^{3}\right )^{{1}/{4}}d \textit {\_a}}{\left (y-a \right )^{{3}/{4}} \sqrt {y-b}}+c_1 &= 0 \\
\int _{}^{y}\frac {1}{\left (\textit {\_a} -a \right )^{{3}/{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} +\frac {i \int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y-b \right )^{2} \left (y-a \right )^{3}\right )^{{1}/{4}}d \textit {\_a}}{\left (y-a \right )^{{3}/{4}} \sqrt {y-b}}+c_1 &= 0 \\
\int _{}^{y}\frac {1}{\left (\textit {\_a} -a \right )^{{3}/{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} -\frac {i \int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y-b \right )^{2} \left (y-a \right )^{3}\right )^{{1}/{4}}d \textit {\_a}}{\left (y-a \right )^{{3}/{4}} \sqrt {y-b}}+c_1 &= 0 \\
\int _{}^{y}\frac {1}{\left (\textit {\_a} -a \right )^{{3}/{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} +\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y-b \right )^{2} \left (y-a \right )^{3}\right )^{{1}/{4}}d \textit {\_a}}{\left (y-a \right )^{{3}/{4}} \sqrt {y-b}}+c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 1.339 (sec). Leaf size: 369
ode=(D[y[x],x])^4 +f[x] (y[x]-a)^3 (y[x]-b)^2==0;
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 14.558 (sec). Leaf size: 221
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
f = Function("f")
ode = Eq((-a + y(x))**3*(-b + y(x))**2*f(x) + Derivative(y(x), x)**4,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - \frac {2 i \sqrt {- b + y{\left (x \right )}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {\left (- b + y{\left (x \right )}\right ) e^{2 i \pi }}{\operatorname {polar\_lift}{\left (a - b \right )}}} \right )}}{\operatorname {polar\_lift}^{\frac {3}{4}}{\left (a - b \right )}} = C_{1} - \int \sqrt [4]{f{\left (x \right )}}\, dx, \ - \frac {2 i \sqrt {- b + y{\left (x \right )}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {\left (- b + y{\left (x \right )}\right ) e^{2 i \pi }}{\operatorname {polar\_lift}{\left (a - b \right )}}} \right )}}{\operatorname {polar\_lift}^{\frac {3}{4}}{\left (a - b \right )}} = C_{1} + \int \sqrt [4]{f{\left (x \right )}}\, dx, \ - \frac {2 i \sqrt {- b + y{\left (x \right )}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {\left (- b + y{\left (x \right )}\right ) e^{2 i \pi }}{\operatorname {polar\_lift}{\left (a - b \right )}}} \right )}}{\operatorname {polar\_lift}^{\frac {3}{4}}{\left (a - b \right )}} = C_{1} - i \int \sqrt [4]{f{\left (x \right )}}\, dx, \ - \frac {2 i \sqrt {- b + y{\left (x \right )}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {\left (- b + y{\left (x \right )}\right ) e^{2 i \pi }}{\operatorname {polar\_lift}{\left (a - b \right )}}} \right )}}{\operatorname {polar\_lift}^{\frac {3}{4}}{\left (a - b \right )}} = C_{1} + i \int \sqrt [4]{f{\left (x \right )}}\, dx\right ]
\]