Internal problem ID [15433]
Internal file name [OUTPUT/15434_Wednesday_May_08_2024_03_58_44_PM_33700203/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 15.5 Linear equations with variable coefficients.
The Lagrange method. Exercises page 148
Problem number: 669.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "kovacic", "exact linear second order ode", "second_order_integrable_as_is", "second_order_ode_missing_y"
Maple gives the following as the ode type
[[_2nd_order, _missing_y]]
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x=\frac {1}{x^{2}+1}} \] With initial conditions \begin {align*} \left [y \left (\infty \right ) = \frac {\pi ^{2}}{8}, y^{\prime }\left (0\right ) = 0\right ] \end {align*}
Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (\left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x \right )d x &= \int \frac {1}{x^{2}+1}d x\\ \left (x^{2}+1\right ) y^{\prime } = \arctan \left (x \right ) + c_{1} \end {align*}
Which is now solved for \(y\). Integrating both sides gives \begin {align*} y &= \int { \frac {\arctan \left (x \right )+c_{1}}{x^{2}+1}\,\mathop {\mathrm {d}x}}\\ &= \frac {\arctan \left (x \right )^{2}}{2}+c_{1} \arctan \left (x \right )+c_{2} \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = \frac {\arctan \left (x \right )^{2}}{2}+c_{1} \arctan \left (x \right )+c_{2} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = \frac {\pi ^{2}}{8}\) and \(x = \infty \) in the above gives \begin {align*} \frac {\pi ^{2}}{8} = \frac {1}{8} \pi ^{2}+\frac {1}{2} c_{1} \pi +c_{2}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = \frac {\arctan \left (x \right )}{x^{2}+1}+\frac {c_{1}}{x^{2}+1} \end {align*}
substituting \(y^{\prime } = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = c_{1}\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=0\\ c_{2}&=0 \end {align*}
Substituting these values back in above solution results in \begin {align*} y = \frac {\arctan \left (x \right )^{2}}{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\arctan \left (x \right )^{2}}{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {\arctan \left (x \right )^{2}}{2} \] Verified OK.
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*} \left (x^{2}+1\right ) p^{\prime }\left (x \right )+2 p \left (x \right ) x -\frac {1}{x^{2}+1} = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode.
Entering Linear first order ODE solver. The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {2 x^{3}+2 x}{\left (x^{2}+1\right )^{2}}d x} \\ &= x^{2}+1 \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu p\right ) &= \left (\mu \right ) \left (\frac {1}{\left (x^{2}+1\right )^{2}}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\left (x^{2}+1\right ) p\right ) &= \left (x^{2}+1\right ) \left (\frac {1}{\left (x^{2}+1\right )^{2}}\right )\\ \mathrm {d} \left (\left (x^{2}+1\right ) p\right ) &= \frac {1}{x^{2}+1}\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \left (x^{2}+1\right ) p &= \int {\frac {1}{x^{2}+1}\,\mathrm {d} x}\\ \left (x^{2}+1\right ) p &= \arctan \left (x \right ) + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =x^{2}+1\) results in \begin {align*} p \left (x \right ) &= \frac {\arctan \left (x \right )}{x^{2}+1}+\frac {c_{1}}{x^{2}+1} \end {align*}
which simplifies to \begin {align*} p \left (x \right ) &= \frac {\arctan \left (x \right )+c_{1}}{x^{2}+1} \end {align*}
Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(p=0\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 0 = c_{1} \end {align*}
The solutions are \begin {align*} c_{1} = 0 \end {align*}
Trying the constant \begin {align*} c_{1} = 0 \end {align*}
Substituting this in the general solution gives \begin {align*} p \left (x \right )&=\frac {\arctan \left (x \right )}{x^{2}+1} \end {align*}
The constant \(c_{1} = 0\) gives valid solution.
Since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = \frac {\arctan \left (x \right )}{x^{2}+1} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { \frac {\arctan \left (x \right )}{x^{2}+1}\,\mathop {\mathrm {d}x}}\\ &= \frac {\arctan \left (x \right )^{2}}{2}+c_{2} \end {align*}
Initial conditions are used to solve for \(c_{2}\). Substituting \(x=\infty \) and \(y=\frac {\pi ^{2}}{8}\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} \frac {\pi ^{2}}{8} = \frac {\pi ^{2}}{8}+c_{2} \end {align*}
The solutions are \begin {align*} c_{2} = 0 \end {align*}
Trying the constant \begin {align*} c_{2} = 0 \end {align*}
Substituting this in the general solution gives \begin {align*} y&=\frac {\arctan \left (x \right )^{2}}{2} \end {align*}
The constant \(c_{2} = 0\) gives valid solution.
Initial conditions are used to solve for the constants of integration.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\arctan \left (x \right )^{2}}{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {\arctan \left (x \right )^{2}}{2} \] Verified OK.
Writing the ode as \[ \left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x = \frac {1}{x^{2}+1} \] Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (\left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x \right )d x &= \int \frac {1}{x^{2}+1}d x\\ y^{\prime }+y^{\prime } x^{2} = \arctan \left (x \right ) +c_{1} \end {align*}
Which is now solved for \(y\). Integrating both sides gives \begin {align*} y &= \int { \frac {\arctan \left (x \right )+c_{1}}{x^{2}+1}\,\mathop {\mathrm {d}x}}\\ &= \frac {\arctan \left (x \right )^{2}}{2}+c_{1} \arctan \left (x \right )+c_{2} \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = \frac {\arctan \left (x \right )^{2}}{2}+c_{1} \arctan \left (x \right )+c_{2} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = \frac {\pi ^{2}}{8}\) and \(x = \infty \) in the above gives \begin {align*} \frac {\pi ^{2}}{8} = \frac {1}{8} \pi ^{2}+\frac {1}{2} c_{1} \pi +c_{2}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = \frac {\arctan \left (x \right )}{x^{2}+1}+\frac {c_{1}}{x^{2}+1} \end {align*}
substituting \(y^{\prime } = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = c_{1}\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=0\\ c_{2}&=0 \end {align*}
Substituting these values back in above solution results in \begin {align*} y = \frac {\arctan \left (x \right )^{2}}{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\arctan \left (x \right )^{2}}{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {\arctan \left (x \right )^{2}}{2} \] Verified OK.
Writing the ode as \begin {align*} \left (x^{2}+1\right ) y^{\prime \prime }+2 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 &= x^{2}+1 \\ B &= 2 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 {1}{\left (x^{2}+1\right )^{2}}\tag {6} \end {align*}
Comparing the above to (5) shows that \begin {align*} s &= 1\\ t &= \left (x^{2}+1\right )^{2} \end {align*}
Therefore eq. (4) becomes \begin {align*} z''(x) &= \left ( \frac {1}{\left (x^{2}+1\right )^{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) \\ &= 4 - 0 \\ &= 4 \end {align*}
The poles of \(r\) in eq. (7) and the order of each pole are determined by solving for the roots of \(t=\left (x^{2}+1\right )^{2}\). There is a pole at \(x=i\) of order \(2\). There is a pole at \(x=-i\) of order \(2\). Since there is no odd order pole larger than \(2\) and the order at \(\infty \) is \(4\) then the necessary conditions for case one are met. Since there is a pole of order \(2\) then necessary conditions for case two are met. Since pole order is not larger than \(2\) and the order at \(\infty \) is \(4\) then the necessary conditions for case three are met. Therefore \begin {align*} L &= [1, 2, 4, 6, 12] \end {align*}
Attempting to find a solution using case \(n=1\).
Looking at poles of order 2. The partial fractions decomposition of \(r\) is \[ r = -\frac {1}{4 \left (x -i\right )^{2}}-\frac {1}{4 \left (x +i\right )^{2}}-\frac {i}{4 \left (x -i\right )}+\frac {i}{4 x +4 i} \] For the pole at \(x=i\) let \(b\) be the coefficient of \(\frac {1}{ \left (x -i\right )^{2}}\) in the partial fractions decomposition of \(r\) given above. Therefore \(b=-{\frac {1}{4}}\). Hence \begin {alignat*} {2} [\sqrt r]_c &= 0 \\ \alpha _c^{+} &= \frac {1}{2} + \sqrt {1+4 b} &&= {\frac {1}{2}}\\ \alpha _c^{-} &= \frac {1}{2} - \sqrt {1+4 b} &&= {\frac {1}{2}} \end {alignat*}
For the pole at \(x=-i\) let \(b\) be the coefficient of \(\frac {1}{ \left (x +i\right )^{2}}\) in the partial fractions decomposition of \(r\) given above. Therefore \(b=-{\frac {1}{4}}\). Hence \begin {alignat*} {2} [\sqrt r]_c &= 0 \\ \alpha _c^{+} &= \frac {1}{2} + \sqrt {1+4 b} &&= {\frac {1}{2}}\\ \alpha _c^{-} &= \frac {1}{2} - \sqrt {1+4 b} &&= {\frac {1}{2}} \end {alignat*}
Since the order of \(r\) at \(\infty \) is \(4 > 2\) then \begin {alignat*} {2} [\sqrt r]_\infty &= 0 \\ \alpha _{\infty }^{+} &= 0 \\ \alpha _{\infty }^{-} &= 1 \end {alignat*}
The following table summarizes the findings so far for poles and for the order of \(r\) at \(\infty \) where \(r\) is \[ r=\frac {1}{\left (x^{2}+1\right )^{2}} \]
| pole \(c\) location | pole order | \([\sqrt r]_c\) | \(\alpha _c^{+}\) | \(\alpha _c^{-}\) |
| \(i\) | \(2\) | \(0\) | \(\frac {1}{2}\) | \(\frac {1}{2}\) |
| \(-i\) | \(2\) | \(0\) | \(\frac {1}{2}\) | \(\frac {1}{2}\) |
| Order of \(r\) at \(\infty \) | \([\sqrt r]_\infty \) | \(\alpha _\infty ^{+}\) | \(\alpha _\infty ^{-}\) |
| \(4\) | \(0\) | \(0\) | \(1\) |
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 \(\alpha _\infty ^{-} = 1\) then \begin {align*} d &= \alpha _\infty ^{-} - \left ( \alpha _{c_1}^{+}+\alpha _{c_2}^{+} \right ) \\ &= 1 - \left ( 1 \right ) \\ &= 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*}
The above gives \begin {align*} \omega &= \left ( (+) [\sqrt r]_{c_1} + \frac { \alpha _{c_1}^{+} }{x- c_1}\right )+\left ( (+) [\sqrt r]_{c_2} + \frac { \alpha _{c_2}^{+} }{x- c_2}\right ) + (-) [\sqrt r]_\infty \\ &= \frac {1}{2 x -2 i}+\frac {1}{2 x +2 i} + (-) \left ( 0 \right ) \\ &= \frac {1}{2 x -2 i}+\frac {1}{2 x +2 i}\\ &= \frac {x}{x^{2}+1} \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 {1}{2 x -2 i}+\frac {1}{2 x +2 i}\right ) \left (0\right ) + \left ( \left (-\frac {1}{2 \left (x -i\right )^{2}}-\frac {1}{2 \left (x +i\right )^{2}}\right ) + \left (\frac {1}{2 x -2 i}+\frac {1}{2 x +2 i}\right )^2 - \left (\frac {1}{\left (x^{2}+1\right )^{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 \left (\frac {1}{2 x -2 i}+\frac {1}{2 x +2 i}\right )d x}\\ &= \sqrt {x^{2}+1} \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 {2 x}{x^{2}+1} \,dx} \\ &= z_1 e^{-\frac {\ln \left (x^{2}+1\right )}{2}} \\ &= z_1 \left (\frac {1}{\sqrt {x^{2}+1}}\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 {2 x}{x^{2}+1} \,dx}}{\left (y_1\right )^2} \,dx \\ &= y_1 \int \frac { e^{-\ln \left (x^{2}+1\right )}}{\left (y_1\right )^2} \,dx \\ &= y_1 \left (\arctan \left (x \right )\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 (\arctan \left (x \right )\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 \[ \left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x = 0 \] The homogeneous solution is found using the Kovacic algorithm which results in \[ y_h = c_{1} +c_{2} \arctan \left (x \right ) \] 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 &= \arctan \left (x \right ) \\ \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 \[ W = \begin {vmatrix} 1 & \arctan \left (x \right ) \\ \frac {d}{dx}\left (1\right ) & \frac {d}{dx}\left (\arctan \left (x \right )\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} 1 & \arctan \left (x \right ) \\ 0 & \frac {1}{x^{2}+1} \end {vmatrix} \] Therefore \[ W = \left (1\right )\left (\frac {1}{x^{2}+1}\right ) - \left (\arctan \left (x \right )\right )\left (0\right ) \] Which simplifies to \[ W = \frac {1}{x^{2}+1} \] Which simplifies to \[ W = \frac {1}{x^{2}+1} \] Therefore Eq. (2) becomes \[ u_1 = -\int \frac {\frac {\arctan \left (x \right )}{x^{2}+1}}{1}\,dx \] Which simplifies to \[ u_1 = - \int \frac {\arctan \left (x \right )}{x^{2}+1}d x \] Hence \[ u_1 = -\frac {\arctan \left (x \right )^{2}}{2} \] And Eq. (3) becomes \[ u_2 = \int \frac {\frac {1}{x^{2}+1}}{1}\,dx \] Which simplifies to \[ u_2 = \int \frac {1}{x^{2}+1}d x \] Hence \[ u_2 = \arctan \left (x \right ) \] Therefore the particular solution, from equation (1) is \[ y_p(x) = \frac {\arctan \left (x \right )^{2}}{2} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} +c_{2} \arctan \left (x \right )\right ) + \left (\frac {\arctan \left (x \right )^{2}}{2}\right ) \\ \end{align*} Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = c_{1} +c_{2} \arctan \left (x \right )+\frac {\arctan \left (x \right )^{2}}{2} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = \frac {\pi ^{2}}{8}\) and \(x = \infty \) in the above gives \begin {align*} \frac {\pi ^{2}}{8} = c_{1} +\frac {1}{2} c_{2} \pi +\frac {1}{8} \pi ^{2}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = \frac {c_{2}}{x^{2}+1}+\frac {\arctan \left (x \right )}{x^{2}+1} \end {align*}
substituting \(y^{\prime } = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = c_{2}\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=0\\ c_{2}&=0 \end {align*}
Substituting these values back in above solution results in \begin {align*} y = \frac {\arctan \left (x \right )^{2}}{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\arctan \left (x \right )^{2}}{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {\arctan \left (x \right )^{2}}{2} \] Verified OK.
An ode of the form \begin {align*} p \left (x \right ) y^{\prime \prime }+q \left (x \right ) y^{\prime }+r \left (x \right ) y&=s \left (x \right ) \end {align*}
is exact if \begin {align*} p''(x) - q'(x) + r(x) &= 0 \tag {1} \end {align*}
For the given ode we have \begin {align*} p(x) &= x^{2}+1\\ q(x) &= 2 x\\ r(x) &= 0\\ s(x) &= \frac {1}{x^{2}+1} \end {align*}
Hence \begin {align*} p''(x) &= 2\\ q'(x) &= 2 \end {align*}
Therefore (1) becomes \begin {align*} 2- \left (2\right ) + \left (0\right )&=0 \end {align*}
Hence the ode is exact. Since we now know the ode is exact, it can be written as \begin {align*} \left (p \left (x \right ) y^{\prime }+\left (q \left (x \right )-p^{\prime }\left (x \right )\right ) y\right )' &= s(x) \end {align*}
Integrating gives \begin {align*} p \left (x \right ) y^{\prime }+\left (q \left (x \right )-p^{\prime }\left (x \right )\right ) y&=\int {s \left (x \right )\, dx} \end {align*}
Substituting the above values for \(p,q,r,s\) gives \begin {align*} \left (x^{2}+1\right ) y^{\prime }&=\int {\frac {1}{x^{2}+1}\, dx} \end {align*}
We now have a first order ode to solve which is \begin {align*} \left (x^{2}+1\right ) y^{\prime } = \arctan \left (x \right )+c_{1} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { \frac {\arctan \left (x \right )+c_{1}}{x^{2}+1}\,\mathop {\mathrm {d}x}}\\ &= \frac {\arctan \left (x \right )^{2}}{2}+c_{1} \arctan \left (x \right )+c_{2} \end {align*}
Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = \frac {\arctan \left (x \right )^{2}}{2}+c_{1} \arctan \left (x \right )+c_{2} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = \frac {\pi ^{2}}{8}\) and \(x = \infty \) in the above gives \begin {align*} \frac {\pi ^{2}}{8} = \frac {1}{8} \pi ^{2}+\frac {1}{2} c_{1} \pi +c_{2}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = \frac {\arctan \left (x \right )}{x^{2}+1}+\frac {c_{1}}{x^{2}+1} \end {align*}
substituting \(y^{\prime } = 0\) and \(x = 0\) in the above gives \begin {align*} 0 = c_{1}\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=0\\ c_{2}&=0 \end {align*}
Substituting these values back in above solution results in \begin {align*} y = \frac {\arctan \left (x \right )^{2}}{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\arctan \left (x \right )^{2}}{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {\arctan \left (x \right )^{2}}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\left (x^{2}+1\right ) \left (\frac {d}{d x}y^{\prime }\right )+2 y^{\prime } x =\frac {1}{x^{2}+1}, y \left (\infty \right )=\frac {\pi ^{2}}{8}, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=0\right ] \\ \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} \\ {} & {} & \left (x^{2}+1\right ) u^{\prime }\left (x \right )+2 u \left (x \right ) x =\frac {1}{x^{2}+1} \\ \bullet & {} & \textrm {Isolate the derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=-\frac {2 x u \left (x \right )}{x^{2}+1}+\frac {1}{\left (x^{2}+1\right )^{2}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} u \left (x \right )\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )+\frac {2 x u \left (x \right )}{x^{2}+1}=\frac {1}{\left (x^{2}+1\right )^{2}} \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (u^{\prime }\left (x \right )+\frac {2 x u \left (x \right )}{x^{2}+1}\right )=\frac {\mu \left (x \right )}{\left (x^{2}+1\right )^{2}} \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d x}\left (u \left (x \right ) \mu \left (x \right )\right ) \\ {} & {} & \mu \left (x \right ) \left (u^{\prime }\left (x \right )+\frac {2 x u \left (x \right )}{x^{2}+1}\right )=u^{\prime }\left (x \right ) \mu \left (x \right )+u \left (x \right ) \mu ^{\prime }\left (x \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (x \right ) \\ {} & {} & \mu ^{\prime }\left (x \right )=\frac {2 \mu \left (x \right ) x}{x^{2}+1} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=x^{2}+1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}\left (u \left (x \right ) \mu \left (x \right )\right )\right )d x =\int \frac {\mu \left (x \right )}{\left (x^{2}+1\right )^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & u \left (x \right ) \mu \left (x \right )=\int \frac {\mu \left (x \right )}{\left (x^{2}+1\right )^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\frac {\int \frac {\mu \left (x \right )}{\left (x^{2}+1\right )^{2}}d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=x^{2}+1 \\ {} & {} & u \left (x \right )=\frac {\int \frac {1}{x^{2}+1}d x +c_{1}}{x^{2}+1} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & u \left (x \right )=\frac {\arctan \left (x \right )+c_{1}}{x^{2}+1} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )=\frac {\arctan \left (x \right )+c_{1}}{x^{2}+1} \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }=\frac {\arctan \left (x \right )+c_{1}}{x^{2}+1} \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int \frac {\arctan \left (x \right )+c_{1}}{x^{2}+1}d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=\frac {\arctan \left (x \right )^{2}}{2}+c_{1} \arctan \left (x \right )+c_{2} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=\frac {\arctan \left (x \right )^{2}}{2}+c_{1} \arctan \left (x \right )+c_{2} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (\infty \right )=\frac {\pi ^{2}}{8} \\ {} & {} & \frac {\pi ^{2}}{8}=\frac {1}{8} \pi ^{2}+\frac {1}{2} c_{1} \pi +c_{2} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\arctan \left (x \right )}{x^{2}+1}+\frac {c_{1}}{x^{2}+1} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =0, c_{2} =0\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {\arctan \left (x \right )^{2}}{2} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {\arctan \left (x \right )^{2}}{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) = -(2*_b(_a)*_a^3+2*_b(_a)*_a-1)/(_a^2+1)^2, _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: 0.063 (sec). Leaf size: 10
dsolve([(1+x^2)*diff(y(x),x$2)+2*x*diff(y(x),x)=1/(1+x^2),y(infinity) = 1/8*Pi^2, D(y)(0) = 0],y(x), singsol=all)
\[ y \left (x \right ) = \frac {\arctan \left (x \right )^{2}}{2} \]
✓ Solution by Mathematica
Time used: 0.044 (sec). Leaf size: 13
DSolve[{(1+x^2)*y''[x]+2*x*y'[x]==1/(1+x^2),{y[Infinity]==Pi^2/8,y'[0]==0}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\arctan (x)^2}{2} \]