Internal problem ID [13245]
Internal file name [OUTPUT/12417_Wednesday_February_14_2024_02_06_06_AM_27663864/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 2. Integration and differential equations. Additional exercises. page
32
Problem number: 2.2 (d).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime } x=\arcsin \left (x^{2}\right )} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {\arcsin \left (x^{2}\right )}{x}\,\mathop {\mathrm {d}x}}\\ &= -\frac {i \arcsin \left (x^{2}\right )^{2}}{4}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1-i x^{2}-\sqrt {-x^{4}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, i x^{2}+\sqrt {-x^{4}+1}\right )}{2}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1+i x^{2}+\sqrt {-x^{4}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -i x^{2}-\sqrt {-x^{4}+1}\right )}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {i \arcsin \left (x^{2}\right )^{2}}{4}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1-i x^{2}-\sqrt {-x^{4}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, i x^{2}+\sqrt {-x^{4}+1}\right )}{2}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1+i x^{2}+\sqrt {-x^{4}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -i x^{2}-\sqrt {-x^{4}+1}\right )}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {i \arcsin \left (x^{2}\right )^{2}}{4}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1-i x^{2}-\sqrt {-x^{4}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, i x^{2}+\sqrt {-x^{4}+1}\right )}{2}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1+i x^{2}+\sqrt {-x^{4}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -i x^{2}-\sqrt {-x^{4}+1}\right )}{2}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x =\arcsin \left (x^{2}\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\arcsin \left (x^{2}\right )}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {\arcsin \left (x^{2}\right )}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {\mathrm {I} \arcsin \left (x^{2}\right )^{2}}{4}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1-\mathrm {I} x^{2}-\sqrt {-x^{4}+1}\right )}{2}-\frac {\mathrm {I} \mathit {polylog}\left (2, \mathrm {I} x^{2}+\sqrt {-x^{4}+1}\right )}{2}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1+\mathrm {I} x^{2}+\sqrt {-x^{4}+1}\right )}{2}-\frac {\mathrm {I} \mathit {polylog}\left (2, -\mathrm {I} x^{2}-\sqrt {-x^{4}+1}\right )}{2}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 88
dsolve(x*diff(y(x),x)=arcsin(x^2),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {i \arcsin \left (x^{2}\right )^{2}}{4}+\frac {\arcsin \left (x^{2}\right ) \ln \left (i x^{2}+\sqrt {-x^{4}+1}+1\right )}{2}-\frac {i \operatorname {polylog}\left (2, \left (i x^{2}+\sqrt {-x^{4}+1}\right )^{2}\right )}{4}+\frac {\arcsin \left (x^{2}\right ) \ln \left (1-i x^{2}-\sqrt {-x^{4}+1}\right )}{2}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 56
DSolve[x*y'[x]==ArcSin[x^2],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {1}{4} i \left (\arcsin \left (x^2\right )^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (x^2\right )}\right )\right )+\frac {1}{2} \arcsin \left (x^2\right ) \log \left (1-e^{2 i \arcsin \left (x^2\right )}\right )+c_1 \]