Internal problem ID [2627]
Internal file name [OUTPUT/2119_Sunday_June_05_2022_02_49_12_AM_27167669/index.tex]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth
edition, 2015
Section: Chapter 1, First-Order Differential Equations. Section 1.4, Separable Differential
Equations. page 43
Problem number: Problem 6.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }-\frac {2 x \left (y-1\right )}{x^{2}+3}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {x \left (2 y -2\right )}{x^{2}+3} \end {align*}
Where \(f(x)=\frac {x}{x^{2}+3}\) and \(g(y)=2 y -2\). Integrating both sides gives \begin{align*} \frac {1}{2 y -2} \,dy &= \frac {x}{x^{2}+3} \,d x \\ \int { \frac {1}{2 y -2} \,dy} &= \int {\frac {x}{x^{2}+3} \,d x} \\ \frac {\ln \left (y -1\right )}{2}&=\frac {\ln \left (x^{2}+3\right )}{2}+c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} \sqrt {y -1} &= {\mathrm e}^{\frac {\ln \left (x^{2}+3\right )}{2}+c_{1}} \end {align*}
Which simplifies to \begin {align*} \sqrt {y -1} &= c_{2} \sqrt {x^{2}+3} \end {align*}
Which simplifies to \[ y = c_{2}^{2} \left (x^{2}+3\right ) {\mathrm e}^{2 c_{1}}+1 \]
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2}^{2} \left (x^{2}+3\right ) {\mathrm e}^{2 c_{1}}+1 \\ \end{align*}
Verification of solutions
\[ y = c_{2}^{2} \left (x^{2}+3\right ) {\mathrm e}^{2 c_{1}}+1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {2 x \left (y-1\right )}{x^{2}+3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {2 x \left (y-1\right )}{x^{2}+3} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y-1}=\frac {2 x}{x^{2}+3} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y-1}d x =\int \frac {2 x}{x^{2}+3}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y-1\right )=\ln \left (x^{2}+3\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{c_{1}} x^{2}+3 \,{\mathrm e}^{c_{1}}+1 \end {array} \]
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: 14
dsolve(diff(y(x),x)=(2*x*(y(x)-1))/(x^2+3),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{2}+3 c_{1} +1 \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 20
DSolve[y'[x]==(2*x*(y[x]-1))/(x^2+3),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 1+c_1 \left (x^2+3\right ) \\ y(x)\to 1 \\ \end{align*}