Internal problem ID [11313]
Internal file name [OUTPUT/10299_Tuesday_December_27_2022_04_05_56_AM_53925476/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 57. Dependent variable absent. Page 132
Problem number: Ex 3.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "kovacic", "second_order_ode_missing_y"
Maple gives the following as the ode type
[[_2nd_order, _missing_y]]
\[ \boxed {y^{\prime \prime }+y^{\prime } x=x} \]
This is second order ode with missing dependent variable \(y\). Let \begin {align*} p(x) &= y^{\prime } \end {align*}
Then \begin {align*} p'(x) &= y^{\prime \prime } \end {align*}
Hence the ode becomes \begin {align*} p^{\prime }\left (x \right )+p \left (x \right ) x -x = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. In canonical form the ODE is \begin {align*} p' &= F(x,p)\\ &= f( x) g(p)\\ &= x \left (-p +1\right ) \end {align*}
Where \(f(x)=x\) and \(g(p)=-p +1\). Integrating both sides gives \begin{align*} \frac {1}{-p +1} \,dp &= x \,d x \\ \int { \frac {1}{-p +1} \,dp} &= \int {x \,d x} \\ -\ln \left (p -1\right )&=\frac {x^{2}}{2}+c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} \frac {1}{p -1} &= {\mathrm e}^{\frac {x^{2}}{2}+c_{1}} \end {align*}
Which simplifies to \begin {align*} \frac {1}{p -1} &= c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \end {align*}
Since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = \frac {\left (c_{2} {\mathrm e}^{\frac {x^{2}}{2}+c_{1}}+1\right ) {\mathrm e}^{-\frac {x^{2}}{2}-c_{1}}}{c_{2}} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { \frac {\left (c_{2} {\mathrm e}^{\frac {x^{2}}{2}+c_{1}}+1\right ) {\mathrm e}^{-\frac {x^{2}}{2}-c_{1}}}{c_{2}}\,\mathop {\mathrm {d}x}}\\ &= x +\frac {{\mathrm e}^{-c_{1}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2 c_{2}}+c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x +\frac {{\mathrm e}^{-c_{1}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2 c_{2}}+c_{3} \\ \end{align*}
Verification of solutions
\[ y = x +\frac {{\mathrm e}^{-c_{1}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2 c_{2}}+c_{3} \] Verified OK.
Writing the ode as \begin {align*} y^{\prime \prime }+y^{\prime } x &= 0 \tag {1} \\ A y^{\prime \prime } + B y^{\prime } + C y &= 0 \tag {2} \end {align*}
Comparing (1) and (2) shows that \begin {align*} A &= 1 \\ B &= x\tag {3} \\ C &= 0 \end {align*}
Applying the Liouville transformation on the dependent variable gives \begin {align*} z(x) &= y e^{\int \frac {B}{2 A} \,dx} \end {align*}
Then (2) becomes \begin {align*} z''(x) = r z(x)\tag {4} \end {align*}
Where \(r\) is given by \begin {align*} r &= \frac {s}{t}\tag {5} \\ &= \frac {2 A B' - 2 B A' + B^2 - 4 A C}{4 A^2} \end {align*}
Substituting the values of \(A,B,C\) from (3) in the above and simplifying gives \begin {align*} r &= \frac {x^{2}+2}{4}\tag {6} \end {align*}
Comparing the above to (5) shows that \begin {align*} s &= x^{2}+2\\ t &= 4 \end {align*}
Therefore eq. (4) becomes \begin {align*} z''(x) &= \left ( \frac {x^{2}}{4}+\frac {1}{2}\right ) z(x)\tag {7} \end {align*}
Equation (7) is now solved. After finding \(z(x)\) then \(y\) is found using the inverse transformation \begin {align*} y &= z \left (x \right ) e^{-\int \frac {B}{2 A} \,dx} \end {align*}
The first step is to determine the case of Kovacic algorithm this ode belongs to. There are 3 cases depending on the order of poles of \(r\) and the order of \(r\) at \(\infty \). The following table summarizes these cases.
| Case |
Allowed pole order for \(r\) |
Allowed value for \(\mathcal {O}(\infty )\) |
| 1 |
\(\left \{ 0,1,2,4,6,8,\cdots \right \} \) |
\(\left \{ \cdots ,-6,-4,-2,0,2,3,4,5,6,\cdots \right \} \) |
|
2 |
Need to have at least one pole that is either order \(2\) or odd order greater than \(2\). Any other pole order is allowed as long as the above condition is satisfied. Hence the following set of pole orders are all allowed. \(\{1,2\}\),\(\{1,3\}\),\(\{2\}\),\(\{3\}\),\(\{3,4\}\),\(\{1,2,5\}\). |
no condition |
| 3 |
\(\left \{ 1,2\right \} \) |
\(\left \{ 2,3,4,5,6,7,\cdots \right \} \) |
The order of \(r\) at \(\infty \) is the degree of \(t\) minus the degree of \(s\). Therefore \begin {align*} O\left (\infty \right ) &= \text {deg}(t) - \text {deg}(s) \\ &= 0 - 2 \\ &= -2 \end {align*}
There are no poles in \(r\). Therefore the set of poles \(\Gamma \) is empty. Since there is no odd order pole larger than \(2\) and the order at \(\infty \) is \(-2\) then the necessary conditions for case one are met. Therefore \begin {align*} L &= [1] \end {align*}
Attempting to find a solution using case \(n=1\).
Since the order of \(r\) at \(\infty \) is \(O_r(\infty ) = -2\) then \begin {alignat*} {3} v &= \frac {-O_r(\infty )}{2} &&= \frac {2}{2} &&= 1 \end {alignat*}
\([\sqrt r]_\infty \) is the sum of terms involving \(x^i\) for \(0\leq i \leq v\) in the Laurent series for \(\sqrt r\) at \(\infty \). Therefore \begin {align*} [\sqrt r]_\infty &= \sum _{i=0}^{v} a_i x^i \\ &= \sum _{i=0}^{1} a_i x^i \tag {8} \end {align*}
Let \(a\) be the coefficient of \(x^v=x^1\) in the above sum. The Laurent series of \(\sqrt r\) at \(\infty \) is \[ \sqrt r \approx \frac {x}{2}+\frac {1}{2 x}-\frac {1}{4 x^{3}}+\frac {1}{4 x^{5}}-\frac {5}{16 x^{7}}+\frac {7}{16 x^{9}}-\frac {21}{32 x^{11}}+\frac {33}{32 x^{13}} + \dots \tag {9} \] Comparing Eq. (9) with Eq. (8) shows that \[ a = {\frac {1}{2}} \] From Eq. (9) the sum up to \(v=1\) gives \begin {align*} [\sqrt r]_\infty &= \sum _{i=0}^{1} a_i x^i \\ &= \frac {x}{2} \tag {10} \end {align*}
Now we need to find \(b\), where \(b\) be the coefficient of \(x^{v-1} = x^{0}=1\) in \(r\) minus the coefficient of same term but in \(\left ( [\sqrt r]_\infty \right )^2 \) where \([\sqrt r]_\infty \) was found above in Eq (10). Hence \[ \left ( [\sqrt r]_\infty \right )^2 = \frac {x^{2}}{4} \] This shows that the coefficient of \(1\) in the above is \(0\). Now we need to find the coefficient of \(1\) in \(r\). How this is done depends on if \(v=0\) or not. Since \(v=1\) which is not zero, then starting \(r=\frac {s}{t}\), we do long division and write this in the form \[ r = Q + \frac {R}{t} \] Where \(Q\) is the quotient and \(R\) is the remainder. Then the coefficient of \(1\) in \(r\) will be the coefficient this term in the quotient. Doing long division gives \begin {align*} r &= \frac {s}{t} \\ &= \frac {x^{2}+2}{4} \\ &= Q + \frac {R}{4} \\ &= \left (\frac {x^{2}}{4}+\frac {1}{2}\right ) + \left ( 0\right ) \\ &= \frac {x^{2}}{4}+\frac {1}{2} \end {align*}
We see that the coefficient of the term \(\frac {1}{x}\) in the quotient is \(\frac {1}{2}\). Now \(b\) can be found. \begin {align*} b &= \left ({\frac {1}{2}}\right )-\left (0\right )\\ &= {\frac {1}{2}} \end {align*}
Hence \begin {alignat*} {3} [\sqrt r]_\infty &= \frac {x}{2}\\ \alpha _{\infty }^{+} &= \frac {1}{2} \left ( \frac {b}{a} - v \right ) &&= \frac {1}{2} \left ( \frac {{\frac {1}{2}}}{{\frac {1}{2}}} - 1 \right ) &&= 0\\ \alpha _{\infty }^{-} &= \frac {1}{2} \left ( -\frac {b}{a} - v \right ) &&= \frac {1}{2} \left ( -\frac {{\frac {1}{2}}}{{\frac {1}{2}}} - 1 \right ) &&= -1 \end {alignat*}
The following table summarizes the findings so far for poles and for the order of \(r\) at \(\infty \) where \(r\) is \[ r=\frac {x^{2}}{4}+\frac {1}{2} \]
| Order of \(r\) at \(\infty \) | \([\sqrt r]_\infty \) | \(\alpha _\infty ^{+}\) | \(\alpha _\infty ^{-}\) |
| \(-2\) | \(\frac {x}{2}\) | \(0\) | \(-1\) |
Now that the all \([\sqrt r]_c\) and its associated \(\alpha _c^{\pm }\) have been determined for all the poles in the set \(\Gamma \) and \([\sqrt r]_\infty \) and its associated \(\alpha _\infty ^{\pm }\) have also been found, the next step is to determine possible non negative integer \(d\) from these using \begin {align*} d &= \alpha _\infty ^{s(\infty )} - \sum _{c \in \Gamma } \alpha _c^{s(c)} \end {align*}
Where \(s(c)\) is either \(+\) or \(-\) and \(s(\infty )\) is the sign of \(\alpha _\infty ^{\pm }\). This is done by trial over all set of families \(s=(s(c))_{c \in \Gamma \cup {\infty }}\) until such \(d\) is found to work in finding candidate \(\omega \). Trying \(\alpha _\infty ^{+} = 0\), and since there are no poles, then \begin {align*} d &= \alpha _\infty ^{+} \\ &= 0 \end {align*}
Since \(d\) an integer and \(d \geq 0\) then it can be used to find \(\omega \) using \begin {align*} \omega &= \sum _{c \in \Gamma } \left ( s(c) [\sqrt r]_c + \frac {\alpha _c^{s(c)}}{x-c} \right ) + s(\infty ) [\sqrt r]_\infty \end {align*}
Substituting the above values in the above results in \begin {align*} \omega &= (+) [\sqrt r]_\infty \\ &= 0 + \left ( \frac {x}{2} \right ) \\ &= \frac {x}{2}\\ &= \frac {x}{2} \end {align*}
Now that \(\omega \) is determined, the next step is find a corresponding minimal polynomial \(p(x)\) of degree \(d=0\) to solve the ode. The polynomial \(p(x)\) needs to satisfy the equation \begin {align*} p'' + 2 \omega p' + \left ( \omega ' +\omega ^2 -r\right ) p = 0 \tag {1A} \end {align*}
Let \begin {align*} p(x) &= 1\tag {2A} \end {align*}
Substituting the above in eq. (1A) gives \begin {align*} \left (0\right ) + 2 \left (\frac {x}{2}\right ) \left (0\right ) + \left ( \left ({\frac {1}{2}}\right ) + \left (\frac {x}{2}\right )^2 - \left (\frac {x^{2}}{4}+\frac {1}{2}\right ) \right ) &= 0\\ 0 = 0 \end {align*}
The equation is satisfied since both sides are zero. Therefore the first solution to the ode \(z'' = r z\) is \begin {align*} z_1(x) &= p e^{ \int \omega \,dx} \\ &= {\mathrm e}^{\int \frac {x}{2}d x}\\ &= {\mathrm e}^{\frac {x^{2}}{4}} \end {align*}
The first solution to the original ode in \(y\) is found from \begin{align*} y_1 &= z_1 e^{ \int -\frac {1}{2} \frac {B}{A} \,dx} \\ &= z_1 e^{ -\int \frac {1}{2} \frac {x}{1} \,dx} \\ &= z_1 e^{-\frac {x^{2}}{4}} \\ &= z_1 \left ({\mathrm e}^{-\frac {x^{2}}{4}}\right ) \\ \end{align*} Which simplifies to \[ y_1 = 1 \] The second solution \(y_2\) to the original ode is found using reduction of order \[ y_2 = y_1 \int \frac { e^{\int -\frac {B}{A} \,dx}}{y_1^2} \,dx \] Substituting gives \begin{align*} y_2 &= y_1 \int \frac { e^{\int -\frac {x}{1} \,dx}}{\left (y_1\right )^2} \,dx \\ &= y_1 \int \frac { e^{-\frac {x^{2}}{2}}}{\left (y_1\right )^2} \,dx \\ &= y_1 \left (\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2}\right ) \\ \end{align*} Therefore the solution is
\begin{align*} y &= c_{1} y_1 + c_{2} y_2 \\ &= c_{1} \left (1\right ) + c_{2} \left (1\left (\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2}\right )\right ) \\ \end{align*} This is second order nonhomogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE \( A y''(x) + B y'(x) + C y(x) = 0\), and \(y_p\) is a particular solution to the nonhomogeneous ODE \(A y''(x) + B y'(x) + C y(x) = f(x)\). \(y_h\) is the solution to \[ y^{\prime \prime }+y^{\prime } x = 0 \] The homogeneous solution is found using the Kovacic algorithm which results in \[ y_h = c_{1} +\frac {c_{2} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2} \] The particular solution \(y_p\) can be found using either the method of undetermined coefficients, or the method of variation of parameters. The method of variation of parameters will be used as it is more general and can be used when the coefficients of the ODE depend on \(x\) as well. Let \begin{equation} \tag{1} y_p(x) = u_1 y_1 + u_2 y_2 \end{equation} Where \(u_1,u_2\) to be determined, and \(y_1,y_2\) are the two basis solutions (the two linearly independent solutions of the homogeneous ODE) found earlier when solving the homogeneous ODE as \begin{align*} y_1 &= 1 \\ y_2 &= \frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2} \\ \end{align*} In the Variation of parameters \(u_1,u_2\) are found using \begin{align*} \tag{2} u_1 &= -\int \frac {y_2 f(x)}{a W(x)} \\ \tag{3} u_2 &= \int \frac {y_1 f(x)}{a W(x)} \\ \end{align*} Where \(W(x)\) is the Wronskian and \(a\) is the coefficient in front of \(y''\) in the given ODE. The Wronskian is given by \(W= \begin {vmatrix} y_1 & y_{2} \\ y_{1}^{\prime } & y_{2}^{\prime } \end {vmatrix} \). Hence \[ W = \begin {vmatrix} 1 & \frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2} \\ \frac {d}{dx}\left (1\right ) & \frac {d}{dx}\left (\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2}\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} 1 & \frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2} \\ 0 & {\mathrm e}^{-\frac {x^{2}}{2}} \end {vmatrix} \] Therefore \[ W = \left (1\right )\left ({\mathrm e}^{-\frac {x^{2}}{2}}\right ) - \left (\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2}\right )\left (0\right ) \] Which simplifies to \[ W = {\mathrm e}^{-\frac {x^{2}}{2}} \] Which simplifies to \[ W = {\mathrm e}^{-\frac {x^{2}}{2}} \] Therefore Eq. (2) becomes \[ u_1 = -\int \frac {\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right ) x}{2}}{{\mathrm e}^{-\frac {x^{2}}{2}}}\,dx \] Which simplifies to \[ u_1 = - \int \frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right ) x \,{\mathrm e}^{\frac {x^{2}}{2}}}{2}d x \] Hence \[ u_1 = -\frac {\sqrt {2}\, \left (-\sqrt {2}\, {\mathrm e}^{-\frac {x^{2}}{2}} {\mathrm e}^{\frac {x^{2}}{2}} x +{\mathrm e}^{\frac {x^{2}}{2}} \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right ) \sqrt {\pi }\right )}{2} \] And Eq. (3) becomes \[ u_2 = \int \frac {x}{{\mathrm e}^{-\frac {x^{2}}{2}}}\,dx \] Which simplifies to \[ u_2 = \int x \,{\mathrm e}^{\frac {x^{2}}{2}}d x \] Hence \[ u_2 = {\mathrm e}^{\frac {x^{2}}{2}} \] Which simplifies to \begin{align*} u_1 &= \frac {\sqrt {2}\, \left (-{\mathrm e}^{\frac {x^{2}}{2}} \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right ) \sqrt {\pi }+x \sqrt {2}\right )}{2} \\ u_2 &= {\mathrm e}^{\frac {x^{2}}{2}} \\ \end{align*} Therefore the particular solution, from equation (1) is \[ y_p(x) = \frac {\sqrt {2}\, \left (-{\mathrm e}^{\frac {x^{2}}{2}} \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right ) \sqrt {\pi }+x \sqrt {2}\right )}{2}+\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{2}}}{2} \] Which simplifies to \[ y_p(x) = x \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} +\frac {c_{2} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2}\right ) + \left (x\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} +\frac {c_{2} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2}+x \\ \end{align*}
Verification of solutions
\[ y = c_{1} +\frac {c_{2} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2}+x \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+y^{\prime } x =x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )+u \left (x \right ) x =x \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=-u \left (x \right ) x +x \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {u^{\prime }\left (x \right )}{u \left (x \right )-1}=-x \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {u^{\prime }\left (x \right )}{u \left (x \right )-1}d x =\int -x d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (u \left (x \right )-1\right )=-\frac {x^{2}}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )={\mathrm e}^{-\frac {x^{2}}{2}+c_{1}}+1 \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )={\mathrm e}^{-\frac {x^{2}}{2}+c_{1}}+1 \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }={\mathrm e}^{-\frac {x^{2}}{2}+c_{1}}+1 \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int \left ({\mathrm e}^{-\frac {x^{2}}{2}+c_{1}}+1\right )d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=x +\frac {{\mathrm e}^{c_{1}} \sqrt {\pi }\, \sqrt {2}\, \mathrm {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2}+c_{2} \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -_b(_a)*_a+_a, _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- high order exact linear fully integrable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 23
dsolve(diff(y(x),x$2)+x*diff(y(x),x)=x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )}{2}+x +c_{2} \]
✓ Solution by Mathematica
Time used: 0.137 (sec). Leaf size: 29
DSolve[y''[x]+x*y'[x]==x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {\frac {\pi }{2}} c_1 \text {erf}\left (\frac {x}{\sqrt {2}}\right )+x+c_2 \]