Internal problem ID [5080]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 2. Linear homogeneous equations. Section 2.3.4 problems. page 104
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {\left (\cos \relax (x )-\sin \relax (x )\right ) y^{\prime \prime }-2 y^{\prime } \sin \relax (x )+\left (\sin \relax (x )+\cos \relax (x )\right ) y-\left (\cos \relax (x )-\sin \relax (x )\right )^{2}=0} \end {gather*}
✗ Solution by Maple
dsolve((cos(x)-sin(x))*diff(y(x),x$2)-2*sin(x)*diff(y(x),x)+(cos(x)+sin(x))*y(x)=(cos(x)-sin(x))^2,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 9.11 (sec). Leaf size: 1077
DSolve[(Cos[x]-Sin[x])*y''[x]-2*Sin[x]*y'[x]+(Cos[x]+Sin[x])*y[x]==(Cos[x]-Sin[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (-1)^{\left (-\frac {1}{8}-\frac {i}{8}\right ) \left (-1+\sqrt {3}\right )} \left (e^{i x}\right )^{\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {3}\right )} \left (i^{\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {3}} c_1 \, _2F_1\left (\frac {1}{2} \left (1+i \sqrt {3}\right ),\frac {1}{2} \left (1+\sqrt {3}\right );\frac {1}{2} \left (2+(1+i) \sqrt {3}\right );-i e^{2 i x}\right ) \left (e^{i x}\right )^{(1+i) \sqrt {3}}+i^{\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {3}} \, _2F_1\left (\frac {1}{2} \left (1+i \sqrt {3}\right ),\frac {1}{2} \left (1+\sqrt {3}\right );\frac {1}{2} \left (2+(1+i) \sqrt {3}\right );-i e^{2 i x}\right ) \int _1^{e^{i x}}-\frac {\left (1-\frac {3 i}{2}\right ) (-1)^{\left (-\frac {1}{8}-\frac {i}{8}\right ) \left (i+\sqrt {3}\right )} \left ((2-i)+\sqrt {3}\right ) \, _2F_1\left (\frac {1}{2}-\frac {\sqrt {3}}{2},\frac {1}{2}-\frac {i \sqrt {3}}{2};1-\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {3};-i K[1]^2\right ) K[1]^{\left (-\frac {1}{2}-\frac {i}{2}\right ) \left ((3-2 i)+\sqrt {3}\right )} \left (K[1]^2-i\right )}{\sqrt {2} \left (\left ((4+i)+\sqrt {3}\right ) \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {3}\right ),\frac {1}{2} \left (1-i \sqrt {3}\right );1+\text {Root}\left [4 \text {$\#$1}^4+9\&,1\right ];-i K[1]^2\right ) \left (1+\text {Root}\left [4 \text {$\#$1}^4+9\&,1\right ]\right ) \, _2F_1\left (\frac {1}{2} \left (1+i \sqrt {3}\right ),\frac {1}{2} \left (1+\sqrt {3}\right );1+\text {Root}\left [4 \text {$\#$1}^4+9\&,4\right ];-i K[1]^2\right )+(5-i) \left (1+\sqrt {3}\right ) \, _2F_1\left (\frac {1}{2} \left (1-\sqrt {3}\right ),\frac {1}{2} \left (1-i \sqrt {3}\right );1+\text {Root}\left [4 \text {$\#$1}^4+9\&,1\right ];-i K[1]^2\right ) \, _2F_1\left (\frac {1}{2} \left (3+i \sqrt {3}\right ),\frac {1}{2} \left (1+\sqrt {3}\right );1+\text {Root}\left [4 \text {$\#$1}^4+9\&,4\right ];-i K[1]^2\right )\right )}dK[1] \left (e^{i x}\right )^{(1+i) \sqrt {3}}+c_2 \, _2F_1\left (\frac {1}{2}-\frac {\sqrt {3}}{2},\frac {1}{2}-\frac {i \sqrt {3}}{2};1-\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {3};-i e^{2 i x}\right )+\, _2F_1\left (\frac {1}{2}-\frac {\sqrt {3}}{2},\frac {1}{2}-\frac {i \sqrt {3}}{2};1-\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {3};-i e^{2 i x}\right ) \int _1^{e^{i x}}\frac {\left (1-\frac {3 i}{2}\right ) (-1)^{\left (\frac {1}{8}+\frac {i}{8}\right ) \left (-i+\sqrt {3}\right )} \left ((2-i)+\sqrt {3}\right ) \, _2F_1\left (\frac {1}{2} \left (1+i \sqrt {3}\right ),\frac {1}{2} \left (1+\sqrt {3}\right );\frac {1}{2} \left (2+(1+i) \sqrt {3}\right );-i K[2]^2\right ) K[2]^{\left (\frac {1}{2}+\frac {i}{2}\right ) \left ((-3+2 i)+\sqrt {3}\right )} \left (K[2]^2-i\right )}{\sqrt {2} \left (\left ((4+i)+\sqrt {3}\right ) \, _2F_1\left (\frac {1}{2} \left (3-\sqrt {3}\right ),\frac {1}{2} \left (1-i \sqrt {3}\right );1+\text {Root}\left [4 \text {$\#$1}^4+9\&,1\right ];-i K[2]^2\right ) \left (1+\text {Root}\left [4 \text {$\#$1}^4+9\&,1\right ]\right ) \, _2F_1\left (\frac {1}{2} \left (1+i \sqrt {3}\right ),\frac {1}{2} \left (1+\sqrt {3}\right );1+\text {Root}\left [4 \text {$\#$1}^4+9\&,4\right ];-i K[2]^2\right )+(5-i) \left (1+\sqrt {3}\right ) \, _2F_1\left (\frac {1}{2} \left (1-\sqrt {3}\right ),\frac {1}{2} \left (1-i \sqrt {3}\right );1+\text {Root}\left [4 \text {$\#$1}^4+9\&,1\right ];-i K[2]^2\right ) \, _2F_1\left (\frac {1}{2} \left (3+i \sqrt {3}\right ),\frac {1}{2} \left (1+\sqrt {3}\right );1+\text {Root}\left [4 \text {$\#$1}^4+9\&,4\right ];-i K[2]^2\right )\right )}dK[2]\right ) \\ \end{align*}