problem 3

Internal problem ID [4763]
Internal ﬁle name [OUTPUT/4256_Sunday_June_05_2022_12_48_43_PM_84879189/index.tex]

Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section: Chapter 8, Ordinary diﬀerential equations. Section 3. Linear First-Order Equations. page 403
Problem number: 3.
ODE order: 1.
ODE degree: 1.

3.3.1 Solving as linear ode

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) &=x \,{\mathrm e}^{-x^{2}} \end {align*}

Hence the ode is \begin {align*} y^{\prime }+2 x y = x \,{\mathrm e}^{-x^{2}} \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}}\left ( \mu y\right ) &= \left (\mu \right ) \left (x \,{\mathrm e}^{-x^{2}}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left ({\mathrm e}^{x^{2}} y\right ) &= \left ({\mathrm e}^{x^{2}}\right ) \left (x \,{\mathrm e}^{-x^{2}}\right )\\ \mathrm {d} \left ({\mathrm e}^{x^{2}} y\right ) &= x\, \mathrm {d} x \end {align*}

Integrating gives \begin {align*} {\mathrm e}^{x^{2}} y &= \int {x\,\mathrm {d} x}\\ {\mathrm e}^{x^{2}} y &= \frac {x^{2}}{2} + c_{1} \end {align*}

Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{x^{2}}\) results in \begin {align*} y &= \frac {x^{2} {\mathrm e}^{-x^{2}}}{2}+c_{1} {\mathrm e}^{-x^{2}} \end {align*}

which simpliﬁes to \begin {align*} y &= {\mathrm e}^{-x^{2}} \left (\frac {x^{2}}{2}+c_{1} \right ) \end {align*}

Summary

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

Veriﬁcation of solutions

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

3.3.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+2 x y=x \,{\mathrm e}^{-x^{2}} \\ \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+x \,{\mathrm e}^{-x^{2}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & y^{\prime }+2 x y=x \,{\mathrm e}^{-x^{2}} \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+2 x y\right )=\mu \left (x \right ) x \,{\mathrm e}^{-x^{2}} \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d x}\left (y \mu \left (x \right )\right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+2 x y\right )=y^{\prime } \mu \left (x \right )+y \mu ^{\prime }\left (x \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (x \right ) \\ {} & {} & \mu ^{\prime }\left (x \right )=2 \mu \left (x \right ) x \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )={\mathrm e}^{x^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}\left (y \mu \left (x \right )\right )\right )d x =\int \mu \left (x \right ) x \,{\mathrm e}^{-x^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int \mu \left (x \right ) x \,{\mathrm e}^{-x^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (x \right ) x \,{\mathrm e}^{-x^{2}}d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )={\mathrm e}^{x^{2}} \\ {} & {} & y=\frac {\int x \,{\mathrm e}^{-x^{2}} {\mathrm e}^{x^{2}}d x +c_{1}}{{\mathrm e}^{x^{2}}} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {\frac {x^{2}}{2}+c_{1}}{{\mathrm e}^{x^{2}}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{-x^{2}} \left (x^{2}+2 c_{1} \right )}{2} \end {array} \]

Maple trace

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

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

3.3 problem 3

3.3.1 Solving as linear ode

3.3.2 Maple step by step solution