1.20 problem 20

Internal problem ID [5733]
Internal file name [OUTPUT/4981_Sunday_June_05_2022_03_15_52_PM_62830200/index.tex]

Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number: 20.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program :

Maple gives the following as the ode type

[_separable]

\[ \boxed {y^{\prime }-\left (y-1\right ) \left (1+x \right )=0} \]

1.20.1 Solving as separable ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \left (y -1\right ) \left (1+x \right ) \end {align*}

Where \(f(x)=1+x\) and \(g(y)=y -1\). Integrating both sides gives \begin{align*} \frac {1}{y -1} \,dy &= 1+x \,d x \\ \int { \frac {1}{y -1} \,dy} &= \int {1+x \,d x} \\ \ln \left (y -1\right )&=\frac {1}{2} x^{2}+x +c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} y -1 &= {\mathrm e}^{\frac {1}{2} x^{2}+x +c_{1}} \end {align*}

Which simplifies to \begin {align*} y -1 &= c_{2} {\mathrm e}^{\frac {1}{2} x^{2}+x} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} {\mathrm e}^{\frac {1}{2} x^{2}+x +c_{1}}+1 \\ \end{align*}

Figure 15: Slope field plot

Verification of solutions

\[ y = c_{2} {\mathrm e}^{\frac {1}{2} x^{2}+x +c_{1}}+1 \] Verified OK.

1.20.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\left (y-1\right ) \left (1+x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (y-1\right ) \left (1+x \right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y-1}=1+x \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y-1}d x =\int \left (1+x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y-1\right )=\frac {1}{2} x^{2}+x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{\frac {1}{2} x^{2}+x +c_{1}}+1 \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 15

dsolve(diff(y(x),x)=(y(x)-1)*(x+1),y(x), singsol=all)
 

\[ y \left (x \right ) = 1+c_{1} {\mathrm e}^{\frac {x \left (x +2\right )}{2}} \]

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 25

DSolve[y'[x]==(y[x]-1)*(x+1),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 1+c_1 e^{\frac {1}{2} x (x+2)} \\ y(x)\to 1 \\ \end{align*}