29.5.23 problem 140

Internal problem ID [4740]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 5
Problem number : 140
Date solved : Monday, January 27, 2025 at 09:35:56 AM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} 2 y^{\prime }+a x&=\sqrt {a^{2} x^{2}-4 b \,x^{2}-4 c y} \end{align*}

✓ Solution by Maple

Time used: 0.086 (sec). Leaf size: 430

dsolve(2*diff(y(x),x)+a*x = sqrt(a^2*x^2-4*b*x^2-4*c*y(x)),y(x), singsol=all)
 

\[ \int _{\textit {\_b}}^{x}\frac {-a \textit {\_a} +\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 c y \left (x \right )}}{a \,\textit {\_a}^{2}-\textit {\_a} \sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 c y \left (x \right )}+4 y \left (x \right )}d \textit {\_a} -2 \left (\int _{}^{y \left (x \right )}\frac {2 \left (\int _{\textit {\_b}}^{x}-\frac {-\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \textit {\_a} a +\left (a^{2}-4 b \right ) \textit {\_a}^{2}-2 \textit {\_f} c}{\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \left (a \,\textit {\_a}^{2}-\textit {\_a} \sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}+4 \textit {\_f} \right )^{2}}d \textit {\_a} \right ) a \,x^{2}-2 \left (\int _{\textit {\_b}}^{x}-\frac {-\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \textit {\_a} a +\left (a^{2}-4 b \right ) \textit {\_a}^{2}-2 \textit {\_f} c}{\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \left (a \,\textit {\_a}^{2}-\textit {\_a} \sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}+4 \textit {\_f} \right )^{2}}d \textit {\_a} \right ) x \sqrt {\left (a^{2}-4 b \right ) x^{2}-4 \textit {\_f} c}+8 \left (\int _{\textit {\_b}}^{x}-\frac {-\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \textit {\_a} a +\left (a^{2}-4 b \right ) \textit {\_a}^{2}-2 \textit {\_f} c}{\sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}\, \left (a \,\textit {\_a}^{2}-\textit {\_a} \sqrt {\left (a^{2}-4 b \right ) \textit {\_a}^{2}-4 \textit {\_f} c}+4 \textit {\_f} \right )^{2}}d \textit {\_a} \right ) \textit {\_f} +1}{a \,x^{2}-x \sqrt {\left (a^{2}-4 b \right ) x^{2}-4 \textit {\_f} c}+4 \textit {\_f}}d \textit {\_f} \right )+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.693 (sec). Leaf size: 542

DSolve[2 D[y[x],x]+a x==Sqrt[a^2 x^2-4 b x^2 -4 c y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^3 c-2 \text {$\#$1}^2 a^2-4 \text {$\#$1}^2 a c+8 \text {$\#$1}^2 b+2 \text {$\#$1} a^2 c-8 \text {$\#$1} b c+a^4-8 a^2 b+16 b^2\&,\frac {\text {$\#$1}^3 \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )+\text {$\#$1}^3 (-\log (x))+\text {$\#$1}^2 c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )-\text {$\#$1}^2 c \log (x)-\text {$\#$1} a^2 \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )+4 \text {$\#$1} b \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )-2 \text {$\#$1} a c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )+a^2 c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )-4 b c \log \left (\text {$\#$1} x-\sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 \sqrt {-c y(x)}\right )+\text {$\#$1} a^2 \log (x)+2 \text {$\#$1} a c \log (x)-4 \text {$\#$1} b \log (x)-a^2 c \log (x)+4 b c \log (x)}{2 \text {$\#$1}^3+3 \text {$\#$1}^2 c-2 \text {$\#$1} a^2-4 \text {$\#$1} a c+8 \text {$\#$1} b+a^2 c-4 b c}\&\right ]-\log \left (\sqrt {-c y(x)} \sqrt {x^2 \left (a^2-4 b\right )-4 c y(x)}+2 c y(x)\right )+\frac {1}{2} \log (y(x))+2 \log (x)=c_1,y(x)\right ] \]