60.3.48 problem 1049

Internal problem ID [11058]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1049
Date solved : Monday, January 27, 2025 at 10:42:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-1\right ) y-{\mathrm e}^{x}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 58

dsolve(diff(diff(y(x),x),x)-4*x*diff(y(x),x)+(4*x^2-1)*y(x)-exp(x)=0,y(x), singsol=all)
 

\[ y = \frac {\left (\sqrt {\pi }\, \left (i \cos \left (x \right )+\sin \left (x \right )\right ) {\mathrm e}^{\frac {i}{2}} \operatorname {erf}\left (x -\frac {1}{2}-\frac {i}{2}\right )-\sqrt {\pi }\, {\mathrm e}^{-\frac {i}{2}} \left (i \cos \left (x \right )-\sin \left (x \right )\right ) \operatorname {erf}\left (x -\frac {1}{2}+\frac {i}{2}\right )+4 \sin \left (x \right ) c_{1} +4 \cos \left (x \right ) c_{2} \right ) {\mathrm e}^{x^{2}}}{4} \]

✓ Solution by Mathematica

Time used: 0.139 (sec). Leaf size: 94

DSolve[-E^x + (-1 + 4*x^2)*y[x] - 4*x*D[y[x],x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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