62.38.12 problem Ex 12

Internal problem ID [13023]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 62. Summary. Page 144
Problem number : Ex 12
Date solved : Tuesday, January 28, 2025 at 04:49:29 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \sin \left (x \right ) y^{\prime \prime }-\cos \left (x \right ) y^{\prime }+2 \sin \left (x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.333 (sec). Leaf size: 39

dsolve(sin(x)*diff(y(x),x$2)-cos(x)*diff(y(x),x)+2*sin(x)*y(x)=0,y(x), singsol=all)
 

\[ y = -\ln \left (\cos \left (x \right )-1\right ) c_{2} \sin \left (x \right )^{2}+\ln \left (\cos \left (x \right )+1\right ) c_{2} \sin \left (x \right )^{2}+\sin \left (x \right )^{2} c_{1} +2 \cos \left (x \right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.276 (sec). Leaf size: 33

DSolve[Sin[x]*D[y[x],{x,2}]-Cos[x]*D[y[x],x]+2*Sin[x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\sin ^2(x) \left (c_2 \int _1^{\cos (x)}\frac {1}{\left (K[1]^2-1\right )^2}dK[1]+c_1\right ) \]