Internal problem ID [6121]
Internal file name [OUTPUT/5369_Sunday_June_05_2022_03_35_35_PM_66222368/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.2 THE NATURE OF SOLUTIONS.
Page 9
Problem number: 2(c).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {\left (1+x \right ) y^{\prime }=x} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {x}{1+x}\,\mathop {\mathrm {d}x}}\\ &= x -\ln \left (1+x \right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x -\ln \left (1+x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = x -\ln \left (1+x \right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (1+x \right ) y^{\prime }=x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x}{1+x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {x}{1+x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=x -\ln \left (1+x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x -\ln \left (1+x \right )+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 13
dsolve((1+x)*diff(y(x),x)=x,y(x), singsol=all)
\[ y \left (x \right ) = x -\ln \left (x +1\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 15
DSolve[(1+x)*y'[x]==x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x-\log (x+1)+c_1 \]