60.2.71 problem 647
Internal
problem
ID
[10658]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
647
Date
solved
:
Tuesday, January 28, 2025 at 05:02:43 PM
CAS
classification
:
[_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\begin{align*} y^{\prime }&=\frac {\left (a y^{2}+b \,x^{2}\right )^{2} x}{a^{{5}/{2}} y} \end{align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 460
dsolve(diff(y(x),x) = (a*y(x)^2+b*x^2)^2*x/a^(5/2)/y(x),y(x), singsol=all)
\begin{align*}
y &= \frac {\sqrt {-a \left (\left (b \,x^{2}-a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}\right ) {\mathrm e}^{\frac {x^{2} \left (-2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}+c_{1} \left (a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right ) {\mathrm e}^{\frac {x^{2} \left (2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}\right ) \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}\right )}}{a \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}\right )} \\
y &= -\frac {\sqrt {-a \left (\left (b \,x^{2}-a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}\right ) {\mathrm e}^{\frac {x^{2} \left (-2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}+c_{1} \left (a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right ) {\mathrm e}^{\frac {x^{2} \left (2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}\right ) \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}\right )}}{a \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 a^{{3}/{2}} \sqrt {-\frac {b}{a^{{3}/{2}}}}+b \,x^{2}\right )}{2 a^{{3}/{2}}}}\right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.347 (sec). Leaf size: 201
DSolve[D[y[x],x] == (x*(b*x^2 + a*y[x]^2)^2)/(a^(5/2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x\left (\frac {a^3 b^2 K[1]}{b^2 K[1]^4+2 a b y(x)^2 K[1]^2+a^2 y(x)^4+a^{3/2} b}-a^{3/2} b K[1]\right )dK[1]+\int _1^{y(x)}\left (\frac {a^4 b K[2]}{b^2 x^4+2 a b K[2]^2 x^2+a^2 K[2]^4+a^{3/2} b}-\int _1^x-\frac {a^3 b^2 K[1] \left (4 a^2 K[2]^3+4 a b K[1]^2 K[2]\right )}{\left (b^2 K[1]^4+2 a b K[2]^2 K[1]^2+a^2 K[2]^4+a^{3/2} b\right )^2}dK[1]\right )dK[2]=c_1,y(x)\right ]
\]