Internal problem ID [6169]
Internal file name [OUTPUT/5417_Sunday_June_05_2022_03_36_43_PM_66708007/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: 2(c).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "exact", "linear", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_linear]
Unable to solve or complete the solution.
\[ \boxed {x \ln \left (x \right ) y^{\prime }+y=3 x^{3}} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}
Where here \begin {align*} p(x) &=\frac {1}{\ln \left (x \right ) x}\\ q(x) &=\frac {3 x^{2}}{\ln \left (x \right )} \end {align*}
Hence the ode is \begin {align*} y^{\prime }+\frac {y}{\ln \left (x \right ) x} = \frac {3 x^{2}}{\ln \left (x \right )} \end {align*}
The domain of \(p(x)=\frac {1}{\ln \left (x \right ) x}\) is \[
\{0
Entering Linear first order ODE solver. The integrating factor \(\mu \) is \begin{align*}
\mu &= {\mathrm e}^{\int \frac {1}{\ln \left (x \right ) x}d x} \\
&= \ln \left (x \right ) \\
\end{align*} The ode becomes
\begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\frac {3 x^{2}}{\ln \left (x \right )}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\ln \left (x \right ) y\right ) &= \left (\ln \left (x \right )\right ) \left (\frac {3 x^{2}}{\ln \left (x \right )}\right )\\ \mathrm {d} \left (\ln \left (x \right ) y\right ) &= \left (3 x^{2}\right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \ln \left (x \right ) y &= \int {3 x^{2}\,\mathrm {d} x}\\ \ln \left (x \right ) y &= x^{3} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\ln \left (x \right )\) results in \begin {align*} y &= \frac {x^{3}}{\ln \left (x \right )}+\frac {c_{1}}{\ln \left (x \right )} \end {align*}
which simplifies to \begin {align*} y &= \frac {x^{3}+c_{1}}{\ln \left (x \right )} \end {align*}
Initial conditions are used to solve for \(c_{1}\). Substituting \(x=1\) and \(y=0\) in the above solution gives an
equation to solve for the constant of integration. Warning: Unable to solve for constant of
integration.
Verification of solutions N/A
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x \ln \left (x \right ) y^{\prime }+y=3 x^{3}, y \left (1\right )=0\right ] \\ \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 {-y+3 x^{3}}{x \ln \left (x \right )} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} y\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y}{\ln \left (x \right ) x}+\frac {3 x^{2}}{\ln \left (x \right )} \\ \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 }+\frac {y}{\ln \left (x \right ) x}=\frac {3 x^{2}}{\ln \left (x \right )} \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+\frac {y}{\ln \left (x \right ) x}\right )=\frac {3 \mu \left (x \right ) x^{2}}{\ln \left (x \right )} \\ \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 }+\frac {y}{\ln \left (x \right ) x}\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 )=\frac {\mu \left (x \right )}{\ln \left (x \right ) x} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\ln \left (x \right ) \\ \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 \frac {3 \mu \left (x \right ) x^{2}}{\ln \left (x \right )}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int \frac {3 \mu \left (x \right ) x^{2}}{\ln \left (x \right )}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \frac {3 \mu \left (x \right ) x^{2}}{\ln \left (x \right )}d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\ln \left (x \right ) \\ {} & {} & y=\frac {\int 3 x^{2}d x +c_{1}}{\ln \left (x \right )} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {x^{3}+c_{1}}{\ln \left (x \right )} \\ \bullet & {} & \textrm {Solution does not satisfy initial condition}\hspace {3pt} \end {array} \]
Maple trace
✗ Solution by Maple
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
Not solved
3.13.2 Solving as linear ode
3.13.3 Maple step by step solution
`Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful`
dsolve([(x*ln(x))*diff(y(x),x)+y(x)=3*x^3,y(1) = 0],y(x), singsol=all)
DSolve[{(x*Log[x])*y'[x]+y[x]==3*x^3,{y[1]==0}},y[x],x,IncludeSingularSolutions -> True]