Internal problem ID [4762]
Internal file name [OUTPUT/4255_Sunday_June_05_2022_12_48_35_PM_350268/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: 2.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {x^{2} y^{\prime }+3 x y=1} \]
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 {3}{x}\\ q(x) &=\frac {1}{x^{2}} \end {align*}
Hence the ode is \begin {align*} y^{\prime }+\frac {3 y}{x} = \frac {1}{x^{2}} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {3}{x}d x} \\ &= x^{3} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\frac {1}{x^{2}}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (x^{3} y\right ) &= \left (x^{3}\right ) \left (\frac {1}{x^{2}}\right )\\ \mathrm {d} \left (x^{3} y\right ) &= x\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} x^{3} y &= \int {x\,\mathrm {d} x}\\ x^{3} y &= \frac {x^{2}}{2} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =x^{3}\) results in \begin {align*} y &= \frac {1}{2 x}+\frac {c_{1}}{x^{3}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{2 x}+\frac {c_{1}}{x^{3}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{2 x}+\frac {c_{1}}{x^{3}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }+3 x y=1 \\ \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 {-3 x y+1}{x^{2}} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} y\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {3 y}{x}+\frac {1}{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 {3 y}{x}=\frac {1}{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 {3 y}{x}\right )=\frac {\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 {3 y}{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 {3 \mu \left (x \right )}{x} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=x^{3} \\ \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 {\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 \frac {\mu \left (x \right )}{x^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \frac {\mu \left (x \right )}{x^{2}}d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=x^{3} \\ {} & {} & y=\frac {\int x d x +c_{1}}{x^{3}} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {\frac {x^{2}}{2}+c_{1}}{x^{3}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=\frac {x^{2}+2 c_{1}}{2 x^{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: 16
dsolve(x^2*diff(y(x),x)+3*x*y(x)=1,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{2}+2 c_{1}}{2 x^{3}} \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 20
DSolve[x^2*y'[x]+3*x*y[x]==1,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^2+2 c_1}{2 x^3} \]