Internal problem ID [9013]
Internal file name [OUTPUT/7948_Monday_June_06_2022_12_59_53_AM_65795571/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 679.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati", "homogeneousTypeD2"
Maple gives the following as the ode type
[[_homogeneous, `class D`], _Riccati]
\[ \boxed {y^{\prime }-\frac {y+x^{3} \ln \left (x \right )+x^{4}+x^{3}+7 x y^{2} \ln \left (x \right )+7 x^{2} y^{2}+7 x y^{2}}{x}=0} \]
Using the change of variables \(y = u \left (x \right ) x\) on the above ode results in new ode in \(u \left (x \right )\) \begin {align*} 7 x^{3} u \left (x \right )^{2} \ln \left (x \right )+7 x^{4} u \left (x \right )^{2}+x^{3} \ln \left (x \right )+x^{4}+7 x^{3} u \left (x \right )^{2}+x^{3}-\left (u^{\prime }\left (x \right ) x +u \left (x \right )\right ) x +u \left (x \right ) x = 0 \end {align*}
In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= x \left (7 u^{2}+1\right ) \left (\ln \left (x \right )+x +1\right ) \end {align*}
Where \(f(x)=\left (\ln \left (x \right )+x +1\right ) x\) and \(g(u)=7 u^{2}+1\). Integrating both sides gives \begin{align*} \frac {1}{7 u^{2}+1} \,du &= \left (\ln \left (x \right )+x +1\right ) x \,d x \\ \int { \frac {1}{7 u^{2}+1} \,du} &= \int {\left (\ln \left (x \right )+x +1\right ) x \,d x} \\ \frac {\sqrt {7}\, \arctan \left (u \sqrt {7}\right )}{7}&=\frac {x^{2} \ln \left (x \right )}{2}+\frac {x^{2}}{4}+\frac {x^{3}}{3}+c_{2} \\ \end{align*} The solution is \[ \frac {\sqrt {7}\, \arctan \left (u \left (x \right ) \sqrt {7}\right )}{7}-\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}-\frac {x^{3}}{3}-c_{2} = 0 \] Replacing \(u(x)\) in the above solution by \(\frac {y}{x}\) results in the solution for \(y\) in implicit form \begin {align*} \frac {\sqrt {7}\, \arctan \left (\frac {y \sqrt {7}}{x}\right )}{7}-\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}-\frac {x^{3}}{3}-c_{2} = 0\\ \frac {\sqrt {7}\, \arctan \left (\frac {y \sqrt {7}}{x}\right )}{7}-\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}-\frac {x^{3}}{3}-c_{2} = 0 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \frac {\sqrt {7}\, \arctan \left (\frac {y \sqrt {7}}{x}\right )}{7}-\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}-\frac {x^{3}}{3}-c_{2} &= 0 \\ \end{align*}
Verification of solutions
\[ \frac {\sqrt {7}\, \arctan \left (\frac {y \sqrt {7}}{x}\right )}{7}-\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}-\frac {x^{3}}{3}-c_{2} = 0 \] Verified OK.
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y +x^{3} \ln \left (x \right )+x^{4}+x^{3}+7 x \,y^{2} \ln \left (x \right )+7 x^{2} y^{2}+7 x \,y^{2}}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = x^{2} \ln \left (x \right )+7 y^{2} \ln \left (x \right )+x^{3}+7 x \,y^{2}+x^{2}+7 y^{2}+\frac {y}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {x^{3} \ln \left (x \right )+x^{4}+x^{3}}{x}\), \(f_1(x)=\frac {1}{x}\) and \(f_2(x)=\frac {7 x \ln \left (x \right )+7 x^{2}+7 x}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {\left (7 x \ln \left (x \right )+7 x^{2}+7 x \right ) u}{x}} \tag {1} \end {align*}
Using the above substitution in the given ODE results (after some simplification)in a second order ODE to solve for \(u(x)\) which is \begin {align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end {align*}
But \begin {align*} f_2' &=\frac {7 \ln \left (x \right )+14+14 x}{x}-\frac {7 x \ln \left (x \right )+7 x^{2}+7 x}{x^{2}}\\ f_1 f_2 &=\frac {7 x \ln \left (x \right )+7 x^{2}+7 x}{x^{2}}\\ f_2^2 f_0 &=\frac {\left (7 x \ln \left (x \right )+7 x^{2}+7 x \right )^{2} \left (x^{3} \ln \left (x \right )+x^{4}+x^{3}\right )}{x^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {\left (7 x \ln \left (x \right )+7 x^{2}+7 x \right ) u^{\prime \prime }\left (x \right )}{x}-\frac {\left (7 \ln \left (x \right )+14+14 x \right ) u^{\prime }\left (x \right )}{x}+\frac {\left (7 x \ln \left (x \right )+7 x^{2}+7 x \right )^{2} \left (x^{3} \ln \left (x \right )+x^{4}+x^{3}\right ) u \left (x \right )}{x^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \sin \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )+c_{2} \cos \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \sqrt {7}\, \left (c_{1} \cos \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )-c_{2} \sin \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )\right ) x \left (\ln \left (x \right )+x +1\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\sqrt {7}\, \left (c_{1} \cos \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )-c_{2} \sin \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )\right ) x^{2} \left (\ln \left (x \right )+x +1\right )}{\left (7 x \ln \left (x \right )+7 x^{2}+7 x \right ) \left (c_{1} \sin \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )+c_{2} \cos \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {x \left (-c_{3} \cos \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )+\sin \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )\right ) \sqrt {7}}{7 c_{3} \sin \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )+7 \cos \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x \left (-c_{3} \cos \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )+\sin \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )\right ) \sqrt {7}}{7 c_{3} \sin \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )+7 \cos \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {x \left (-c_{3} \cos \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )+\sin \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )\right ) \sqrt {7}}{7 c_{3} \sin \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )+7 \cos \left (\frac {\sqrt {7}\, x^{2} \left (6 \ln \left (x \right )+3+4 x \right )}{12}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 7 x y^{2} \ln \left (x \right )+7 x^{2} y^{2}+x^{3} \ln \left (x \right )+x^{4}+7 x y^{2}+x^{3}-y^{\prime } x +y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-y-x^{3} \ln \left (x \right )-x^{4}-x^{3}-7 x y^{2} \ln \left (x \right )-7 x^{2} y^{2}-7 x y^{2}}{x} \end {array} \]
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 homogeneous D <- homogeneous successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 37
dsolve(diff(y(x),x) = (y(x)+x^3*ln(x)+x^4+x^3+7*x*y(x)^2*ln(x)+7*x^2*y(x)^2+7*x*y(x)^2)/x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\tan \left (\frac {\left (6 x^{2} \ln \left (x \right )+4 x^{3}+3 x^{2}+12 c_{1} \right ) \sqrt {7}}{12}\right ) x \sqrt {7}}{7} \]
✓ Solution by Mathematica
Time used: 0.419 (sec). Leaf size: 44
DSolve[y'[x] == (x^3 + x^4 + x^3*Log[x] + y[x] + 7*x*y[x]^2 + 7*x^2*y[x]^2 + 7*x*Log[x]*y[x]^2)/x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x \tan \left (\frac {1}{12} \sqrt {7} \left (4 x^3+3 x^2+6 x^2 \log (x)+12 c_1\right )\right )}{\sqrt {7}} \]