Internal problem ID [11345]
Internal file name [OUTPUT/10332_Friday_January_27_2023_02_37_11_AM_81805463/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IX, Miscellaneous methods for solving equations of higher order than first.
Article 62. Summary. Page 144
Problem number: Ex 11.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_missing_y"
Maple gives the following as the ode type
[[_3rd_order, _missing_y]]
\[ \boxed {4 x^{2} y^{\prime \prime \prime }+8 x y^{\prime \prime }+y^{\prime }=0} \] Since \(y\) is missing from the ode then we can use the substitution \(y^{\prime } = v \left (x \right )\) to reduce the order by one. The ODE becomes \begin {align*} 4 v^{\prime \prime }\left (x \right ) x^{2}+8 v^{\prime }\left (x \right ) x +v \left (x \right ) = 0 \end {align*}
This is Euler second order ODE. Let the solution be \(v \left (x \right ) = x^r\), then \(v'=r x^{r-1}\) and \(v''=r(r-1) x^{r-2}\). Substituting these back into the given ODE gives \[ 4 x^{2}(r(r-1))x^{r-2}+8 x r x^{r-1}+x^{r} = 0 \] Simplifying gives \[ 4 r \left (r -1\right )x^{r}+8 r\,x^{r}+x^{r} = 0 \] Since \(x^{r}\neq 0\) then dividing throughout by \(x^{r}\) gives \[ 4 r \left (r -1\right )+8 r+1 = 0 \] Or \[ 4 r^{2}+4 r +1 = 0 \tag {1} \] Equation (1) is the characteristic equation. Its roots determine the form of the general solution. Using the quadratic equation the roots are \begin {align*} r_1 &= -{\frac {1}{2}}\\ r_2 &= -{\frac {1}{2}} \end {align*}
Since the roots are equal, then the general solution is \[ v \left (x \right )= c_{1} v_1 + c_{2} v_2 \] Where \(v_1 = x^{r}\) and \(v_2 = x^{r} \ln \left (x \right )\). Hence \[ v \left (x \right ) = \frac {c_{1}}{\sqrt {x}}+\frac {c_{2} \ln \left (x \right )}{\sqrt {x}} \] But since \(y^{\prime } = v \left (x \right )\) then we now need to solve the ode \(y^{\prime } = \frac {c_{1}}{\sqrt {x}}+\frac {c_{2} \ln \left (x \right )}{\sqrt {x}}\). Integrating both sides gives \begin {align*} y &= \int { \frac {c_{2} \ln \left (x \right )+c_{1}}{\sqrt {x}}\,\mathop {\mathrm {d}x}}\\ &= 2 \ln \left (x \right ) \sqrt {x}\, c_{2} +2 \sqrt {x}\, c_{1} -4 \sqrt {x}\, c_{2} +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \ln \left (x \right ) \sqrt {x}\, c_{2} +2 \sqrt {x}\, c_{1} -4 \sqrt {x}\, c_{2} +c_{3} \\ \end{align*}
Verification of solutions
\[ y = 2 \ln \left (x \right ) \sqrt {x}\, c_{2} +2 \sqrt {x}\, c_{1} -4 \sqrt {x}\, c_{2} +c_{3} \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type <- LODE of Euler type successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 16
dsolve(4*x^2*diff(y(x),x$3)+8*x*diff(y(x),x$2)+diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{3} \ln \left (x \right )+c_{2} \right ) \sqrt {x}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.05 (sec). Leaf size: 28
DSolve[4*x^2*y'''[x]+8*x*y''[x]+y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {x} (c_2 \log (x)+2 c_1-2 c_2)+c_3 \]