60.1.568 problem 571
Internal
problem
ID
[10582]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
571
Date
solved
:
Monday, January 27, 2025 at 09:16:41 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} a \,x^{n} f \left (y^{\prime }\right )+x y^{\prime }-y&=0 \end{align*}
✓ Solution by Maple
Time used: 1.479 (sec). Leaf size: 169
dsolve(a*x^n*f(diff(y(x),x))+x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\[
\left [y \left (\textit {\_T} \right ) = a {\left ({\left (\frac {\left (-n +1\right ) \left (\int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} \right )+c_{1} a n}{a n f \left (\textit {\_T} \right )}\right )}^{\frac {1}{n -1}} f \left (\textit {\_T} \right )^{\frac {1}{n \left (n -1\right )}}\right )}^{n} f \left (\textit {\_T} \right )+\textit {\_T} {\left (\frac {\left (-n +1\right ) \left (\int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} \right )+c_{1} a n}{a n f \left (\textit {\_T} \right )}\right )}^{\frac {1}{n -1}} f \left (\textit {\_T} \right )^{\frac {1}{n \left (n -1\right )}}, x \left (\textit {\_T} \right ) = {\left (\frac {\left (-n +1\right ) \left (\int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} \right )+c_{1} a n}{a n f \left (\textit {\_T} \right )}\right )}^{\frac {1}{n -1}} f \left (\textit {\_T} \right )^{\frac {1}{n \left (n -1\right )}}\right ]
\]
✓ Solution by Mathematica
Time used: 0.109 (sec). Leaf size: 124
DSolve[a*x^n*f[D[y[x],x]] - y[x] + x*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left \{y(x)=a x^n f(K[1])+x K[1],x=\left (n f(K[1])^{\frac {1}{n}-1} \int _1^{K[1]}-\frac {f(K[2])^{\frac {n-1}{n}-1}}{a n}dK[2]-f(K[1])^{\frac {1}{n}-1} \int _1^{K[1]}-\frac {f(K[2])^{\frac {n-1}{n}-1}}{a n}dK[2]+c_1 f(K[1])^{\frac {1}{n}-1}\right ){}^{\frac {1}{n-1}}\right \},\{y(x),K[1]\}\right ]
\]