60.2.265 problem 841
Internal
problem
ID
[10852]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
841
Date
solved
:
Tuesday, January 28, 2025 at 05:24:25 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{{3}/{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{{3}/{2}} b \,x^{2}+a^{{5}/{2}} y^{4}}{a \,x^{2} y} \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 97
dsolve(diff(y(x),x) = (b*x^3+c^2*a^(1/2)-2*c*b*x^2*a^(1/2)+2*c*y(x)^2*a^(3/2)+b^2*x^4*a^(1/2)-2*y(x)^2*a^(3/2)*b*x^2+a^(5/2)*y(x)^4)/a/x^2/y(x),y(x), singsol=all)
\begin{align*}
y &= -\frac {2 \sqrt {\left (\left (c_{1} x +1\right ) \left (b \,x^{2}-c \right ) \sqrt {a}+\frac {x}{2}\right ) a^{{3}/{2}} \left (c_{1} x +1\right )}}{a^{{3}/{2}} \left (2 c_{1} x +2\right )} \\
y &= \frac {\sqrt {\left (\left (c_{1} x +1\right ) \left (b \,x^{2}-c \right ) \sqrt {a}+\frac {x}{2}\right ) a^{{3}/{2}} \left (c_{1} x +1\right )}}{a^{{3}/{2}} \left (c_{1} x +1\right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.863 (sec). Leaf size: 431
DSolve[D[y[x],x] == (Sqrt[a]*c^2 - 2*Sqrt[a]*b*c*x^2 + b*x^3 + Sqrt[a]*b^2*x^4 + 2*a^(3/2)*c*y[x]^2 - 2*a^(3/2)*b*x^2*y[x]^2 + a^(5/2)*y[x]^4)/(a*x^2*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (\frac {4 a^4 x^2 K[2]}{\left (-2 \sqrt {a} b x^2-x+2 a^{3/2} K[2]^2+2 \sqrt {a} c\right )^2}-\int _1^x\left (\frac {2 K[2] a^{7/2}}{b \left (2 \sqrt {a} b K[1]^2+K[1]-2 a^{3/2} K[2]^2-2 \sqrt {a} c\right )^2}+\frac {4 K[2] \left (-8 b K[1] K[2]^2 a^4+2 K[2]^2 a^{7/2}-8 b c K[1] a^3+2 c a^{5/2}-K[1] a^2\right ) a^{3/2}}{b \left (2 \sqrt {a} b K[1]^2+K[1]-2 a^{3/2} K[2]^2-2 \sqrt {a} c\right )^3}+\frac {4 a^{7/2} K[2]-16 a^4 b K[1] K[2]}{2 b \left (2 \sqrt {a} b K[1]^2+K[1]-2 a^{3/2} K[2]^2-2 \sqrt {a} c\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\left (-a^{5/2}+\frac {a^2}{2 b \left (2 \sqrt {a} b K[1]^2+K[1]-2 a^{3/2} y(x)^2-2 \sqrt {a} c\right )}+\frac {-8 b K[1] y(x)^2 a^4+2 y(x)^2 a^{7/2}-8 b c K[1] a^3+2 c a^{5/2}-K[1] a^2}{2 b \left (2 \sqrt {a} b K[1]^2+K[1]-2 a^{3/2} y(x)^2-2 \sqrt {a} c\right )^2}\right )dK[1]=c_1,y(x)\right ]
\]