44.5.34 problem 34

Internal problem ID [7096]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 34
Date solved : Tuesday, January 28, 2025 at 08:57:25 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.220 (sec). Leaf size: 56

dsolve([diff(y(x),x)=exp(-2*y(x))*sin(x)/(1+x^2),y(0) = 0],y(x), singsol=all)
 

\[ y = \frac {\ln \left (i \sinh \left (1\right ) \pi -2 i \operatorname {Si}\left (i\right ) \cosh \left (1\right )-i \operatorname {Si}\left (-i+x \right ) \cosh \left (1\right )+i \cosh \left (1\right ) \operatorname {Si}\left (x +i\right )-2 \,\operatorname {Ci}\left (i\right ) \sinh \left (1\right )+\operatorname {Ci}\left (-i+x \right ) \sinh \left (1\right )+\operatorname {Ci}\left (x +i\right ) \sinh \left (1\right )+1\right )}{2} \]

✓ Solution by Mathematica

Time used: 1.119 (sec). Leaf size: 123

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

\[ y(x)\to \frac {1}{2} \log \left (\frac {\left (e^2-1\right ) \operatorname {CosIntegral}(i-x)+\left (e^2-1\right ) \operatorname {CosIntegral}(x+i)-2 e^2 \operatorname {CosIntegral}(i)+2 \operatorname {CosIntegral}(i)+2 e^2 \text {Shi}(1)+2 \text {Shi}(1)+i e^2 \text {Si}(i-x)+i \text {Si}(i-x)+i e^2 \text {Si}(x+i)+i \text {Si}(x+i)+2 e}{2 e}\right ) \]