60.2.3 problem 579
Internal
problem
ID
[10590]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
579
Date
solved
:
Monday, January 27, 2025 at 09:16:57 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} y^{\prime }&=-\frac {a x}{2}+F \left (y+\frac {a \,x^{2}}{4}+\frac {b x}{2}\right ) \end{align*}
✓ Solution by Maple
Time used: 0.042 (sec). Leaf size: 35
dsolve(diff(y(x),x) = -1/2*a*x+F(y(x)+1/4*a*x^2+1/2*b*x),y(x), singsol=all)
\[
y = -\frac {a \,x^{2}}{4}-\frac {b x}{2}+\operatorname {RootOf}\left (-x +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 F \left (\textit {\_a} \right )+b}d \textit {\_a} \right )+c_{1} \right )
\]
✓ Solution by Mathematica
Time used: 0.181 (sec). Leaf size: 514
DSolve[D[y[x],x] == -1/2*(a*x) + F[(b*x)/2 + (a*x^2)/4 + y[x]],y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}-\frac {b \int _1^x\left (\frac {2 a K[1] F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}+\frac {2 F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}-\frac {4 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right ) F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}\right )dK[1]+2 F\left (\frac {a x^2}{4}+\frac {b x}{2}+K[2]\right ) \int _1^x\left (\frac {2 a K[1] F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}+\frac {2 F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}-\frac {4 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right ) F''\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}\right )dK[1]+2}{b+2 F\left (\frac {a x^2}{4}+\frac {b x}{2}+K[2]\right )}dK[2]+\int _1^x\left (\frac {2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}-\frac {a K[1]}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}\right )dK[1]=c_1,y(x)\right ]
\]