60.1.312 problem 313
Internal
problem
ID
[10326]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
313
Date
solved
:
Tuesday, January 28, 2025 at 04:36:12 PM
CAS
classification
:
[_rational]
\begin{align*} \left (2 a y^{3}+3 a x y^{2}-b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 b \,x^{2} y+2 b \,x^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.010 (sec). Leaf size: 591
dsolve((2*a*y(x)^3+3*a*x*y(x)^2-b*x^3+c*x^2)*diff(y(x),x)-a*y(x)^3+c*y(x)^2+3*b*x^2*y(x)+2*b*x^3 = 0,y(x), singsol=all)
\begin{align*}
y &= -\frac {\left (-{\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{{2}/{3}}+\left (c x -c_{1} \right ) a 12^{{1}/{3}}\right ) 12^{{1}/{3}}}{6 {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{{1}/{3}} a} \\
y &= -\frac {3^{{1}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{{2}/{3}}+a \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) 2^{{2}/{3}} \left (c x -c_{1} \right )\right ) 2^{{2}/{3}}}{12 {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{{1}/{3}} a} \\
y &= \frac {3^{{1}/{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{{2}/{3}}+\left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) a 2^{{2}/{3}} \left (c x -c_{1} \right )\right ) 2^{{2}/{3}}}{12 {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{{1}/{3}} a} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.338 (sec). Leaf size: 164
DSolve[2*b*x^3 + 3*b*x^2*y[x] + c*y[x]^2 - a*y[x]^3 + (c*x^2 - b*x^3 + 3*a*x*y[x]^2 + 2*a*y[x]^3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (a x-2 a K[2]-\int _1^x\left (b+\frac {3 a K[2]^2-3 b K[2]^2-2 c K[2]}{(K[1]+K[2])^2}-\frac {2 \left (a K[2]^3-b K[2]^3-c K[2]^2\right )}{(K[1]+K[2])^3}\right )dK[1]+\frac {-a x^3+b x^3-c x^2}{(x+K[2])^2}\right )dK[2]+\int _1^x\left (-2 b K[1]+b y(x)+\frac {a y(x)^3-b y(x)^3-c y(x)^2}{(K[1]+y(x))^2}\right )dK[1]=c_1,y(x)\right ]
\]