7.4.30 problem 30

Internal problem ID [102]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.5 (linear equations). Problems at page 54
Problem number : 30
Date solved : Friday, February 07, 2025 at 07:49:28 AM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (1\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 36

dsolve([2*x*diff(y(x),x)=y(x)+2*x*cos(x),y(1) = 0],y(x), singsol=all)
 

\[ y = \sqrt {x}\, \sqrt {2}\, \sqrt {\pi }\, \left (\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right )-\operatorname {FresnelC}\left (\frac {\sqrt {2}}{\sqrt {\pi }}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 149

DSolve[{2*x*D[y[x],x]==y[x]+2*x*Cos[x],{y[1]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {\pi } \left ((-1-i) x \text {erf}\left (\sqrt {i x}\right )+\sqrt {2} \text {erf}\left (\sqrt [4]{-1}\right ) \sqrt {i x} \sqrt {x}-(1+i) x \text {erfi}\left (\sqrt {i x}\right )+\sqrt {2} \text {erfi}\left (\sqrt [4]{-1}\right ) \sqrt {i x} \sqrt {x}-(1-i) \sqrt {x^2}+(1+i) x-(1+i) \sqrt {2} \sqrt {i x} \sqrt {x}\right )}{\sqrt {i x}} \]