77.1.149 problem 176 (page 265)

Internal problem ID [18039]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 176 (page 265)
Date solved : Tuesday, January 28, 2025 at 11:23:21 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.125 (sec). Leaf size: 44

dsolve([diff(y(x),x)=y(x)+z(x),diff(z(x),x)=y(x)+z(x)+x],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.202 (sec). Leaf size: 84

DSolve[{D[y[x],x]==y[x]+z[x],D[z[x],x]==y[x]+z[x]+x},{y[x],z[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{8} \left (-2 x^2-2 x+4 c_1 \left (e^{2 x}+1\right )+4 c_2 e^{2 x}-1-4 c_2\right ) \\ z(x)\to \frac {1}{8} \left (2 x^2-2 x+4 c_1 \left (e^{2 x}-1\right )+4 c_2 e^{2 x}-1+4 c_2\right ) \\ \end{align*}