Internal problem ID [7872]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 292.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class C], _rational]
Solve \begin {gather*} \boxed {\left (a y+x b +c \right )^{2} y^{\prime }+\left (\alpha y+\beta x +\gamma \right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 125
dsolve((a*y(x)+b*x+c)^2*diff(y(x),x)+(alpha*y(x)+beta*x+gamma)^2=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {-b \gamma +\beta c +\RootOf \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2} a^{2}-2 \textit {\_a} a b +b^{2}}{\textit {\_a}^{3} a^{2}-2 \textit {\_a}^{2} a b -\textit {\_a}^{2} \alpha ^{2}+2 \textit {\_a} \alpha \beta +\textit {\_a} \,b^{2}-\beta ^{2}}d \textit {\_a} +\ln \left (x \left (a \beta -b \alpha \right )+a \gamma -\alpha c \right )+c_{1}\right ) \left (x \left (a \beta -b \alpha \right )+a \gamma -\alpha c \right )}{-a \beta +b \alpha } \]
✓ Solution by Mathematica
Time used: 36.646 (sec). Leaf size: 1653
DSolve[(a*y[x]+b*x+c)^2*y'[x]+(\[Alpha]*y[x]+\[Beta]*x+\[Gamma])^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\beta (b \alpha -a \beta ) \text {RootSum}\left [-\gamma ^3 b^3-\alpha ^3 y(x)^3 b^3-\gamma \text {$\#$1}^2 b^3-3 \alpha ^2 \gamma y(x)^2 b^3+2 \alpha ^2 \text {$\#$1} y(x)^2 b^3+2 \gamma ^2 \text {$\#$1} b^3-3 \alpha \gamma ^2 y(x) b^3-\alpha \text {$\#$1}^2 y(x) b^3+4 \alpha \gamma \text {$\#$1} y(x) b^3+3 a \alpha ^2 \beta y(x)^3 b^2+3 c \beta \gamma ^2 b^2+c \beta \text {$\#$1}^2 b^2+3 c \alpha ^2 \beta y(x)^2 b^2+6 a \alpha \beta \gamma y(x)^2 b^2-4 a \alpha \beta \text {$\#$1} y(x)^2 b^2-4 c \beta \gamma \text {$\#$1} b^2+3 a \beta \gamma ^2 y(x) b^2+a \beta \text {$\#$1}^2 y(x) b^2+6 c \alpha \beta \gamma y(x) b^2-4 c \alpha \beta \text {$\#$1} y(x) b^2-4 a \beta \gamma \text {$\#$1} y(x) b^2-\alpha \beta \text {$\#$1}^3 b-3 a^2 \alpha \beta ^2 y(x)^3 b+\alpha \beta \gamma \text {$\#$1}^2 b-6 a c \alpha \beta ^2 y(x)^2 b-3 a^2 \beta ^2 \gamma y(x)^2 b+2 a^2 \beta ^2 \text {$\#$1} y(x)^2 b-3 c^2 \beta ^2 \gamma b+2 c^2 \beta ^2 \text {$\#$1} b-3 c^2 \alpha \beta ^2 y(x) b+\alpha ^2 \beta \text {$\#$1}^2 y(x) b-6 a c \beta ^2 \gamma y(x) b+4 a c \beta ^2 \text {$\#$1} y(x) b+c^3 \beta ^3+a \beta ^2 \text {$\#$1}^3+a^3 \beta ^3 y(x)^3-c \alpha \beta ^2 \text {$\#$1}^2+3 a^2 c \beta ^3 y(x)^2+3 a c^2 \beta ^3 y(x)-a \alpha \beta ^2 \text {$\#$1}^2 y(x)\&,\frac {\log (x \beta +\gamma -\text {$\#$1}+\alpha y(x)) \text {$\#$1}^2}{-2 \gamma ^2 b^3-2 \alpha ^2 y(x)^2 b^3+2 \gamma \text {$\#$1} b^3-4 \alpha \gamma y(x) b^3+2 \alpha \text {$\#$1} y(x) b^3+4 a \alpha \beta y(x)^2 b^2+4 c \beta \gamma b^2-2 c \beta \text {$\#$1} b^2+4 c \alpha \beta y(x) b^2+4 a \beta \gamma y(x) b^2-2 a \beta \text {$\#$1} y(x) b^2-2 c^2 \beta ^2 b+3 \alpha \beta \text {$\#$1}^2 b-2 a^2 \beta ^2 y(x)^2 b-2 \alpha \beta \gamma \text {$\#$1} b-4 a c \beta ^2 y(x) b-2 \alpha ^2 \beta \text {$\#$1} y(x) b-3 a \beta ^2 \text {$\#$1}^2+2 c \alpha \beta ^2 \text {$\#$1}+2 a \alpha \beta ^2 \text {$\#$1} y(x)}\&\right ]+\int _1^{y(x)}\left (\frac {-\beta K[1]^2 a^3+b \alpha K[1]^2 a^2-2 c \beta K[1] a^2-2 b x \beta K[1] a^2-c^2 \beta a-b^2 x^2 \beta a-2 b c x \beta a+2 b c \alpha K[1] a+2 b^2 x \alpha K[1] a+b c^2 \alpha +b^3 x^2 \alpha +2 b^2 c x \alpha }{x^2 \gamma b^3+x^2 \alpha K[1] b^3+2 a x \alpha K[1]^2 b^2-c x^2 \beta b^2+2 c x \gamma b^2+2 c x \alpha K[1] b^2-a x^2 \beta K[1] b^2+2 a x \gamma K[1] b^2+a^2 \alpha K[1]^3 b+x^3 \alpha \beta ^2 b+x \alpha \gamma ^2 b+x \alpha ^3 K[1]^2 b+2 a c \alpha K[1]^2 b-2 a^2 x \beta K[1]^2 b+a^2 \gamma K[1]^2 b-2 c^2 x \beta b+c^2 \gamma b+2 x^2 \alpha \beta \gamma b+c^2 \alpha K[1] b+2 x^2 \alpha ^2 \beta K[1] b-4 a c x \beta K[1] b+2 x \alpha ^2 \gamma K[1] b+2 a c \gamma K[1] b-a x^3 \beta ^3-a \gamma ^3-a^3 \beta K[1]^3+c x^2 \alpha \beta ^2+c \alpha \gamma ^2-3 a x \beta \gamma ^2+c \alpha ^3 K[1]^2-a x \alpha ^2 \beta K[1]^2-3 a^2 c \beta K[1]^2-a \alpha ^2 \gamma K[1]^2-c^3 \beta -3 a x^2 \beta ^2 \gamma +2 c x \alpha \beta \gamma -2 a x^2 \alpha \beta ^2 K[1]-2 a \alpha \gamma ^2 K[1]-3 a c^2 \beta K[1]+2 c x \alpha ^2 \beta K[1]+2 c \alpha ^2 \gamma K[1]-4 a x \alpha \beta \gamma K[1]}-\frac {(a \beta -b \alpha ) \left (c^2+2 b x c+2 a K[1] c+b^2 x^2+a^2 K[1]^2+2 a b x K[1]\right )}{-x^2 \gamma b^3-x^2 \alpha K[1] b^3-2 a x \alpha K[1]^2 b^2+c x^2 \beta b^2-2 c x \gamma b^2-2 c x \alpha K[1] b^2+a x^2 \beta K[1] b^2-2 a x \gamma K[1] b^2-a^2 \alpha K[1]^3 b-x^3 \alpha \beta ^2 b-x \alpha \gamma ^2 b-x \alpha ^3 K[1]^2 b-2 a c \alpha K[1]^2 b+2 a^2 x \beta K[1]^2 b-a^2 \gamma K[1]^2 b+2 c^2 x \beta b-c^2 \gamma b-2 x^2 \alpha \beta \gamma b-c^2 \alpha K[1] b-2 x^2 \alpha ^2 \beta K[1] b+4 a c x \beta K[1] b-2 x \alpha ^2 \gamma K[1] b-2 a c \gamma K[1] b+a x^3 \beta ^3+a \gamma ^3+a^3 \beta K[1]^3-c x^2 \alpha \beta ^2-c \alpha \gamma ^2+3 a x \beta \gamma ^2-c \alpha ^3 K[1]^2+a x \alpha ^2 \beta K[1]^2+3 a^2 c \beta K[1]^2+a \alpha ^2 \gamma K[1]^2+c^3 \beta +3 a x^2 \beta ^2 \gamma -2 c x \alpha \beta \gamma +2 a x^2 \alpha \beta ^2 K[1]+2 a \alpha \gamma ^2 K[1]+3 a c^2 \beta K[1]-2 c x \alpha ^2 \beta K[1]-2 c \alpha ^2 \gamma K[1]+4 a x \alpha \beta \gamma K[1]}\right )dK[1]=c_1,y(x)\right ] \]