Internal problem ID [4774]
Internal file name [OUTPUT/4267_Sunday_June_05_2022_12_50_22_PM_1604250/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: 14.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {x^{\prime }-\frac {3 y^{\frac {2}{3}}-x}{3 y}=0} \]
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} x^{\prime } + p(y)x &= q(y) \end {align*}
Where here \begin {align*} p(y) &=\frac {1}{3 y}\\ q(y) &=\frac {1}{y^{\frac {1}{3}}} \end {align*}
Hence the ode is \begin {align*} x^{\prime }+\frac {x}{3 y} = \frac {1}{y^{\frac {1}{3}}} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {1}{3 y}d y} \\ &= y^{\frac {1}{3}} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}y}}\left ( \mu x\right ) &= \left (\mu \right ) \left (\frac {1}{y^{\frac {1}{3}}}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}y}} \left (x \,y^{\frac {1}{3}}\right ) &= \left (y^{\frac {1}{3}}\right ) \left (\frac {1}{y^{\frac {1}{3}}}\right )\\ \mathrm {d} \left (x \,y^{\frac {1}{3}}\right ) &= \mathrm {d} y \end {align*}
Integrating gives \begin {align*} x \,y^{\frac {1}{3}} &= \int {\mathrm {d} y}\\ x \,y^{\frac {1}{3}} &= y + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =y^{\frac {1}{3}}\) results in \begin {align*} x &= y^{\frac {2}{3}}+\frac {c_{1}}{y^{\frac {1}{3}}} \end {align*}
which simplifies to \begin {align*} x &= \frac {y +c_{1}}{y^{\frac {1}{3}}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= \frac {y +c_{1}}{y^{\frac {1}{3}}} \\ \end{align*}
Verification of solutions
\[ x = \frac {y +c_{1}}{y^{\frac {1}{3}}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }-\frac {3 y^{\frac {2}{3}}-x}{3 y}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=\frac {3 y^{\frac {2}{3}}-x}{3 y} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} x\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & x^{\prime }=-\frac {x}{3 y}+\frac {1}{y^{\frac {1}{3}}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} x\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & x^{\prime }+\frac {x}{3 y}=\frac {1}{y^{\frac {1}{3}}} \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (y \right ) \\ {} & {} & \mu \left (y \right ) \left (x^{\prime }+\frac {x}{3 y}\right )=\frac {\mu \left (y \right )}{y^{\frac {1}{3}}} \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d y}\left (x \mu \left (y \right )\right ) \\ {} & {} & \mu \left (y \right ) \left (x^{\prime }+\frac {x}{3 y}\right )=x^{\prime } \mu \left (y \right )+x \mu ^{\prime }\left (y \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (y \right ) \\ {} & {} & \mu ^{\prime }\left (y \right )=\frac {\mu \left (y \right )}{3 y} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (y \right )=y^{\frac {1}{3}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} y \\ {} & {} & \int \left (\frac {d}{d y}\left (x \mu \left (y \right )\right )\right )d y =\int \frac {\mu \left (y \right )}{y^{\frac {1}{3}}}d y +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & x \mu \left (y \right )=\int \frac {\mu \left (y \right )}{y^{\frac {1}{3}}}d y +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=\frac {\int \frac {\mu \left (y \right )}{y^{\frac {1}{3}}}d y +c_{1}}{\mu \left (y \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (y \right )=y^{\frac {1}{3}} \\ {} & {} & x=\frac {\int 1d y +c_{1}}{y^{\frac {1}{3}}} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & x=\frac {y +c_{1}}{y^{\frac {1}{3}}} \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: 11
dsolve(diff(x(y),y)=(3*y^(2/3)-x(y))/(3*y),x(y), singsol=all)
\[ x \left (y \right ) = \frac {y +c_{1}}{y^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 15
DSolve[x'[y]==(3*y^(2/3)-x[y])/(3*y),x[y],y,IncludeSingularSolutions -> True]
\[ x(y)\to \frac {y+c_1}{\sqrt [3]{y}} \]