1.1 problem 1

Internal problem ID [5075]
Internal file name [OUTPUT/4568_Sunday_June_05_2022_03_01_02_PM_43400769/index.tex]

Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section: Program 24. First order differential equations. Test excercise 24. page 1067
Problem number: 1.
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]

\[ \boxed {y^{\prime } x=x^{2}+2 x -3} \]

1.1.1 Solving as quadrature ode

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

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{2}}{2}+2 x -3 \ln \left (x \right )+c_{1} \\ \end{align*}

Figure 1: Slope field plot

Verification of solutions

\[ y = \frac {x^{2}}{2}+2 x -3 \ln \left (x \right )+c_{1} \] Verified OK.

1.1.2 Maple step by step solution

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

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: 18

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

\[ y \left (x \right ) = \frac {x^{2}}{2}+2 x -3 \ln \left (x \right )+c_{1} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 22

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

\[ y(x)\to \frac {x^2}{2}+2 x-3 \log (x)+c_1 \]