Internal problem ID [9296]
Internal file name [OUTPUT/8232_Monday_June_06_2022_02_24_37_AM_30025390/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 963.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]
\[ \boxed {y^{\prime }-\frac {-4 \cos \left (x \right ) x +4 \sin \left (x \right ) x^{2}+4 x +4+4 y^{2}+8 y \cos \left (x \right ) x -8 x y+2 x^{2} \cos \left (2 x \right )+6 x^{2}-8 x^{2} \cos \left (x \right )+4 y^{3}+12 y^{2} \cos \left (x \right ) x -12 x y^{2}+6 y x^{2} \cos \left (2 x \right )+18 x^{2} y-24 y \cos \left (x \right ) x^{2}+x^{3} \cos \left (3 x \right )+15 x^{3} \cos \left (x \right )-6 x^{3} \cos \left (2 x \right )-10 x^{3}}{4 x}=0} \]
This is Abel first kind ODE, it has the form \[ y^{\prime }= f_0(x)+f_1(x) y +f_2(x)y^{2}+f_3(x)y^{3} \] Comparing the above to given ODE which is \begin {align*} y^{\prime }&=\frac {y^{3}}{x}+\frac {\left (12 \cos \left (x \right ) x -12 x +4\right ) y^{2}}{4 x}+\frac {\left (6 x^{2} \cos \left (2 x \right )-24 x^{2} \cos \left (x \right )+8 \cos \left (x \right ) x +18 x^{2}-8 x \right ) y}{4 x}+\frac {-6 x^{3} \cos \left (2 x \right )+x^{3} \cos \left (3 x \right )+15 x^{3} \cos \left (x \right )+2 x^{2} \cos \left (2 x \right )-8 x^{2} \cos \left (x \right )+4 \sin \left (x \right ) x^{2}-10 x^{3}-4 \cos \left (x \right ) x +6 x^{2}+4 x +4}{4 x}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= -3 x^{2} \cos \left (x \right )^{2}-x^{2}+\cos \left (x \right )^{3} x^{2}+3 x^{2} \cos \left (x \right )+\cos \left (x \right )^{2} x +x -2 \cos \left (x \right ) x +x \sin \left (x \right )-\cos \left (x \right )+1+\frac {1}{x}\\ f_1(x) &= 3 \cos \left (x \right )^{2} x +3 x -6 \cos \left (x \right ) x +2 \cos \left (x \right )-2\\ f_2(x) &= 3 \cos \left (x \right )-3+\frac {1}{x}\\ f_3(x) &= \frac {1}{x} \end {align*}
Since \(f_2(x)=3 \cos \left (x \right )-3+\frac {1}{x}\) is not zero, then the first step is to apply the following transformation to remove \(f_2\). Let \(y = u(x) - \frac {f_2}{3 f_3}\) or \begin {align*} y &= u(x) - \left ( \frac {3 \cos \left (x \right )-3+\frac {1}{x}}{\frac {3}{x}} \right ) \\ &= -\cos \left (x \right ) x +u \left (x \right )+x -\frac {1}{3} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = \frac {u \left (x \right )^{3}}{x}-\frac {u \left (x \right )}{3 x}+\frac {29}{27 x}\tag {2} \end {align*}
The above ODE (2) can now be solved as separable.
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= \frac {u^{3}-\frac {1}{3} u +\frac {29}{27}}{x} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(u)=u^{3}-\frac {1}{3} u +\frac {29}{27}\). Integrating both sides gives \begin{align*} \frac {1}{u^{3}-\frac {1}{3} u +\frac {29}{27}} \,du &= \frac {1}{x} \,d x \\ \int { \frac {1}{u^{3}-\frac {1}{3} u +\frac {29}{27}} \,du} &= \int {\frac {1}{x} \,d x} \\ \int _{}^{u}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a}&=\ln \left (x \right )+c_{2} \\ \end{align*} Which results in \[ \int _{}^{u}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a}=\ln \left (x \right )+c_{2} \] The solution is \[ \int _{}^{u \left (x \right )}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} = 0 \] Substituting \(u=y-\frac {\left (3 \cos \left (x \right )-3+\frac {1}{x}\right ) x}{3}\) in the above solution gives \begin {align*} \int _{}^{y-\frac {\left (3 \cos \left (x \right )-3+\frac {1}{x}\right ) x}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} = 0 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y-\frac {\left (3 \cos \left (x \right )-3+\frac {1}{x}\right ) x}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} &= 0 \\ \end{align*}
Verification of solutions
\[ \int _{}^{y-\frac {\left (3 \cos \left (x \right )-3+\frac {1}{x}\right ) x}{3}}\frac {1}{\textit {\_a}^{3}-\frac {1}{3} \textit {\_a} +\frac {29}{27}}d \textit {\_a} -\ln \left (x \right )-c_{2} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 6 y x^{2} \cos \left (2 x \right )-6 x^{3} \cos \left (2 x \right )+x^{3} \cos \left (3 x \right )+12 y^{2} \cos \left (x \right ) x -24 y \cos \left (x \right ) x^{2}+15 x^{3} \cos \left (x \right )+2 x^{2} \cos \left (2 x \right )+8 y \cos \left (x \right ) x -8 x^{2} \cos \left (x \right )+4 \sin \left (x \right ) x^{2}+4 y^{3}-12 x y^{2}+18 x^{2} y-10 x^{3}-4 \cos \left (x \right ) x -4 y^{\prime } x +4 y^{2}-8 x y+6 x^{2}+4 x +4=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {4 \cos \left (x \right ) x -4 \sin \left (x \right ) x^{2}-4 x -4-4 y^{2}-8 y \cos \left (x \right ) x +8 x y-2 x^{2} \cos \left (2 x \right )-6 x^{2}+8 x^{2} \cos \left (x \right )-4 y^{3}-12 y^{2} \cos \left (x \right ) x +12 x y^{2}-6 y x^{2} \cos \left (2 x \right )-18 x^{2} y+24 y \cos \left (x \right ) x^{2}-x^{3} \cos \left (3 x \right )-15 x^{3} \cos \left (x \right )+6 x^{3} \cos \left (2 x \right )+10 x^{3}}{4 x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 39
dsolve(diff(y(x),x) = 1/4*(-4*cos(x)*x+4*sin(x)*x^2+4*x+4+4*y(x)^2+8*y(x)*cos(x)*x-8*x*y(x)+2*x^2*cos(2*x)+6*x^2-8*x^2*cos(x)+4*y(x)^3+12*y(x)^2*cos(x)*x-12*x*y(x)^2+6*y(x)*x^2*cos(2*x)+18*x^2*y(x)-24*y(x)*cos(x)*x^2+x^3*cos(3*x)+15*x^3*cos(x)-6*x^3*cos(2*x)-10*x^3)/x,y(x), singsol=all)
\[ y \left (x \right ) = -\cos \left (x \right ) x +x -\frac {1}{3}+\frac {29 \operatorname {RootOf}\left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+\ln \left (x \right )+3 c_{1} \right )}{9} \]
✓ Solution by Mathematica
Time used: 0.375 (sec). Leaf size: 108
DSolve[y'[x] == (1 + x + (3*x^2)/2 - (5*x^3)/2 - x*Cos[x] - 2*x^2*Cos[x] + (15*x^3*Cos[x])/4 + (x^2*Cos[2*x])/2 - (3*x^3*Cos[2*x])/2 + (x^3*Cos[3*x])/4 + x^2*Sin[x] - 2*x*y[x] + (9*x^2*y[x])/2 + 2*x*Cos[x]*y[x] - 6*x^2*Cos[x]*y[x] + (3*x^2*Cos[2*x]*y[x])/2 + y[x]^2 - 3*x*y[x]^2 + 3*x*Cos[x]*y[x]^2 + y[x]^3)/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {29}{3} \text {RootSum}\left [-29 \text {$\#$1}^3+3 \sqrt [3]{29} \text {$\#$1}-29\&,\frac {\log \left (\frac {\frac {3 y(x)}{x}+\frac {-3 x+3 x \cos (x)+1}{x}}{\sqrt [3]{29} \sqrt [3]{\frac {1}{x^3}}}-\text {$\#$1}\right )}{\sqrt [3]{29}-29 \text {$\#$1}^2}\&\right ]=\frac {1}{9} 29^{2/3} \left (\frac {1}{x^3}\right )^{2/3} x^2 \log (x)+c_1,y(x)\right ] \]