29.3.18 problem 72

Internal problem ID [4680]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 3
Problem number : 72
Date solved : Monday, January 27, 2025 at 09:31:19 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }+4 \csc \left (x \right )&=\left (3-\cot \left (x \right )\right ) y+y^{2} \sin \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 66

dsolve(diff(y(x),x)+4*csc(x) = (3-cot(x))*y(x)+y(x)^2*sin(x),y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {3 \csc \left (x \right ) \left (c_{1} \left (\operatorname {csgn}\left (\sin \left (x \right )\right )+\frac {5}{3}\right ) \left (\cos \left (x \right )+i \sin \left (x \right )\right )^{-\frac {5 i}{2}}+\left (\cos \left (x \right )+i \sin \left (x \right )\right )^{\frac {5 i}{2}} \left (\operatorname {csgn}\left (\sin \left (x \right )\right )-\frac {5}{3}\right )\right )}{2 c_{1} \left (\cos \left (x \right )+i \sin \left (x \right )\right )^{-\frac {5 i}{2}}+2 \left (\cos \left (x \right )+i \sin \left (x \right )\right )^{\frac {5 i}{2}}} \]

✓ Solution by Mathematica

Time used: 0.250 (sec). Leaf size: 32

DSolve[D[y[x],x]+4 Csc[x]==(3-Cot[x])y[x]+y[x]^2 Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \left (-4+\frac {1}{\frac {1}{5}+c_1 e^{5 x}}\right ) \csc (x) \\ y(x)\to -4 \csc (x) \\ \end{align*}