Internal problem ID [6162]
Internal file name [OUTPUT/5410_Sunday_June_05_2022_03_36_32_PM_80153340/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 1.4 First Order Linear Equations.
Page 15
Problem number: 1(f).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }+2 x y=0} \]
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}
Where here \begin {align*} p(x) &=2 x\\ q(x) &=0 \end {align*}
Hence the ode is \begin {align*} y^{\prime }+2 x y = 0 \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int 2 x d x} \\ &= {\mathrm e}^{x^{2}} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \mu y &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left ({\mathrm e}^{x^{2}} y\right ) &= 0 \end {align*}
Integrating gives \begin {align*} {\mathrm e}^{x^{2}} y &= c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{x^{2}}\) results in \begin {align*} y &= c_{1} {\mathrm e}^{-x^{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-x^{2}} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-x^{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+2 x y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-2 x y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-2 x \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -2 x d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-x^{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{-x^{2}+c_{1}} \end {array} \]
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: 12
dsolve(diff(y(x),x)+2*x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-x^{2}} c_{1} \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 20
DSolve[y'[x]+2*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^{-x^2} \\ y(x)\to 0 \\ \end{align*}