Internal problem ID [5079]
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: 11.
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 } \cos \relax (x )+\left (\cos \relax (x )-\sin \relax (x )\right ) y-\left (\cos \relax (x )+\sin \relax (x )\right )^{2} {\mathrm e}^{2 x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.625 (sec). Leaf size: 477
dsolve((cos(x)+sin(x))*diff(y(x),x$2)-2*cos(x)*diff(y(x),x)+(cos(x)-sin(x))*y(x)=(cos(x)+sin(x))^2*exp(2*x),y(x), singsol=all)
\[ y \relax (x ) = \cos \relax (x ) c_{2}+\cos \relax (x ) \left (\int -{\mathrm e}^{\int \frac {-4 \left (\cos ^{4}\relax (x )\right )+2 \sin \relax (x ) \left (\cos ^{3}\relax (x )\right )+5 \left (\cos ^{2}\relax (x )\right )-2}{2 \sin \relax (x ) \left (\cos ^{3}\relax (x )\right )-\sin \relax (x ) \cos \relax (x )}d x} \sin \relax (x )d x \right ) c_{1}+\cos \relax (x ) \left (\left (\int \frac {\left (\cos \relax (x )+\sin \relax (x )\right ) {\mathrm e}^{2 x +4 \left (\int \frac {\cos ^{3}\relax (x )}{2 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )-\sin \relax (x )}d x \right )-2 \left (\int \frac {\cos ^{2}\relax (x )}{2 \left (\cos ^{2}\relax (x )\right )-1}d x \right )-5 \left (\int \frac {\cos \relax (x )}{2 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )-\sin \relax (x )}d x \right )+2 \left (\int \frac {1}{\sin \relax (x ) \cos \relax (x ) \left (2 \left (\cos ^{2}\relax (x )\right )-1\right )}d x \right )}}{\sin \relax (x ) \cos \relax (x )}d x \right ) \left (\int {\mathrm e}^{-4 \left (\int \frac {\cos ^{3}\relax (x )}{2 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )-\sin \relax (x )}d x \right )+2 \left (\int \frac {\cos ^{2}\relax (x )}{2 \left (\cos ^{2}\relax (x )\right )-1}d x \right )+5 \left (\int \frac {\cos \relax (x )}{2 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )-\sin \relax (x )}d x \right )-2 \left (\int \frac {1}{\sin \relax (x ) \cos \relax (x ) \left (2 \left (\cos ^{2}\relax (x )\right )-1\right )}d x \right )} \sin \relax (x )d x \right )-\left (\int \frac {\left (\cos \relax (x )+\sin \relax (x )\right ) \left (\int {\mathrm e}^{-4 \left (\int \frac {\cos ^{3}\relax (x )}{2 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )-\sin \relax (x )}d x \right )+2 \left (\int \frac {\cos ^{2}\relax (x )}{2 \left (\cos ^{2}\relax (x )\right )-1}d x \right )+5 \left (\int \frac {\cos \relax (x )}{2 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )-\sin \relax (x )}d x \right )-2 \left (\int \frac {1}{\sin \relax (x ) \cos \relax (x ) \left (2 \left (\cos ^{2}\relax (x )\right )-1\right )}d x \right )} \sin \relax (x )d x \right ) {\mathrm e}^{2 x +4 \left (\int \frac {\cos ^{3}\relax (x )}{2 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )-\sin \relax (x )}d x \right )-2 \left (\int \frac {\cos ^{2}\relax (x )}{2 \left (\cos ^{2}\relax (x )\right )-1}d x \right )-5 \left (\int \frac {\cos \relax (x )}{2 \left (\cos ^{2}\relax (x )\right ) \sin \relax (x )-\sin \relax (x )}d x \right )+2 \left (\int \frac {1}{\sin \relax (x ) \cos \relax (x ) \left (2 \left (\cos ^{2}\relax (x )\right )-1\right )}d x \right )}}{\sin \relax (x ) \cos \relax (x )}d x \right )\right ) \]
✓ Solution by Mathematica
Time used: 3.183 (sec). Leaf size: 337
DSolve[(Cos[x]+Sin[x])*y''[x]-2*Cos[x]*y'[x]+(Cos[x]-Sin[x])*y[x]==(Cos[x]+Sin[x])^2*Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (e^{i x}\right )^{-1-2 i} \left (e^{6 i x} \sqrt {1+e^{-4 i x}}+e^{8 i x} \sqrt {1+e^{-4 i x}}+e^{10 i x} \sqrt {1+e^{-4 i x}}+(1+i) \sqrt {-e^{4 i x} \left (1+e^{4 i x}\right )} \sqrt {-\left (1+e^{4 i x}\right )^2} \left (1+(2-2 i) c_1 \left (e^{i x}\right )^{1+i}\right )+(1-i) e^{2 i x} \sqrt {-\left (1+e^{4 i x}\right )^2} \left (\sqrt {-e^{4 i x} \left (1+e^{4 i x}\right )}+2 c_2 \left (e^{i x}\right )^{2 i} \sqrt {-1-e^{4 i x}}\right )+e^{4 i x} \left (\sqrt {1+e^{-4 i x}}+(2-2 i) c_2 \left (e^{i x}\right )^{2 i} \sqrt {-1-e^{4 i x}} \sqrt {-\left (1+e^{4 i x}\right )^2}\right )\right )}{4 \sqrt {-e^{4 i x} \left (1+e^{4 i x}\right )} \sqrt {-\left (1+e^{4 i x}\right )^2}} \\ \end{align*}