Internal problem ID [2646]
Internal file name [OUTPUT/2138_Sunday_June_05_2022_02_50_01_AM_1763986/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 1.6, First-Order Linear Differential
Equations. page 59
Problem number: Problem 8.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {y^{\prime }+\frac {y}{\ln \left (x \right ) x}=9 x^{2}} \]
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) &=\frac {1}{\ln \left (x \right ) x}\\ q(x) &=9 x^{2} \end {align*}
Hence the ode is \begin {align*} y^{\prime }+\frac {y}{\ln \left (x \right ) x} = 9 x^{2} \end {align*}
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 (9 x^{2}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\ln \left (x \right ) y\right ) &= \left (\ln \left (x \right )\right ) \left (9 x^{2}\right )\\ \mathrm {d} \left (\ln \left (x \right ) y\right ) &= \left (9 x^{2} \ln \left (x \right )\right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \ln \left (x \right ) y &= \int {9 x^{2} \ln \left (x \right )\,\mathrm {d} x}\\ \ln \left (x \right ) y &= 3 x^{3} \ln \left (x \right )-x^{3} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\ln \left (x \right )\) results in \begin {align*} y &= \frac {3 x^{3} \ln \left (x \right )-x^{3}}{\ln \left (x \right )}+\frac {c_{1}}{\ln \left (x \right )} \end {align*}
which simplifies to \begin {align*} y &= \frac {3 x^{3} \ln \left (x \right )-x^{3}+c_{1}}{\ln \left (x \right )} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {3 x^{3} \ln \left (x \right )-x^{3}+c_{1}}{\ln \left (x \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {3 x^{3} \ln \left (x \right )-x^{3}+c_{1}}{\ln \left (x \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\frac {y}{\ln \left (x \right ) x}=9 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 }=-\frac {y}{\ln \left (x \right ) x}+9 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 }+\frac {y}{\ln \left (x \right ) x}=9 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 }+\frac {y}{\ln \left (x \right ) x}\right )=9 \mu \left (x \right ) 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 }+\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 9 \mu \left (x \right ) x^{2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int 9 \mu \left (x \right ) x^{2}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int 9 \mu \left (x \right ) x^{2}d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\ln \left (x \right ) \\ {} & {} & y=\frac {\int 9 x^{2} \ln \left (x \right )d x +c_{1}}{\ln \left (x \right )} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {3 x^{3} \ln \left (x \right )-x^{3}+c_{1}}{\ln \left (x \right )} \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: 23
dsolve(diff(y(x),x)+1/(x*ln(x))*y(x)=9*x^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {3 x^{3} \ln \left (x \right )-x^{3}+c_{1}}{\ln \left (x \right )} \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 25
DSolve[y'[x]+1/(x*Log[x])*y[x]==9*x^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {-x^3+3 x^3 \log (x)+c_1}{\log (x)} \]