1.24 problem 2(j)

Internal problem ID [6128]
Internal file name [OUTPUT/5376_Sunday_June_05_2022_03_35_43_PM_51844411/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(j).
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]

\[ \boxed {\left (x^{2}-3 x +2\right ) y^{\prime }=x} \]

1.24.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { \frac {x}{x^{2}-3 x +2}\,\mathop {\mathrm {d}x}}\\ &= 2 \ln \left (x -2\right )-\ln \left (x -1\right )+c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \ln \left (x -2\right )-\ln \left (x -1\right )+c_{1} \\ \end{align*}

Figure 41: Slope field plot

Verification of solutions

\[ y = 2 \ln \left (x -2\right )-\ln \left (x -1\right )+c_{1} \] Verified OK.

1.24.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{2}-3 x +2\right ) y^{\prime }=x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x}{x^{2}-3 x +2} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {x}{x^{2}-3 x +2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=2 \ln \left (x -2\right )-\ln \left (x -1\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=2 \ln \left (x -2\right )-\ln \left (x -1\right )+c_{1} \end {array} \]

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: 18

dsolve((x^2-3*x+2)*diff(y(x),x)=x,y(x), singsol=all)
 

\[ y \left (x \right ) = 2 \ln \left (-2+x \right )-\ln \left (x -1\right )+c_{1} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 24

DSolve[(x^2-3*x+2)*y'[x]==x,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\log (1-x)+2 \log (2-x)+c_1 \]