Internal problem ID [3107]
Internal file name [OUTPUT/2599_Sunday_June_05_2022_03_21_53_AM_63742245/index.tex]
Book: Differential equations with applications and historial notes, George F. Simmons,
1971
Section: Chapter 2, section 10, page 47
Problem number: 4(e).
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`], _rational, _Riccati]
\[ \boxed {y^{\prime } x -y-9 y^{2}=x^{2}} \]
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*} \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right ) x -u \left (x \right ) x -9 u \left (x \right )^{2} x^{2} = x^{2} \end {align*}
Integrating both sides gives \begin {align*} \int \frac {1}{9 u^{2}+1}d u &= x +c_{2}\\ \frac {\arctan \left (3 u \right )}{3}&=x +c_{2} \end {align*}
Solving for \(u\) gives these solutions \begin {align*} u_1&=\frac {\tan \left (3 x +3 c_{2} \right )}{3} \end {align*}
Therefore the solution \(y\) is \begin {align*} y&=x u\\ &=\frac {x \tan \left (3 x +3 c_{2} \right )}{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x \tan \left (3 x +3 c_{2} \right )}{3} \\ \end{align*}
Verification of solutions
\[ y = \frac {x \tan \left (3 x +3 c_{2} \right )}{3} \] Verified OK.
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {x^{2}+9 y^{2}+y}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = x +\frac {9 y^{2}}{x}+\frac {y}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=x\), \(f_1(x)=\frac {1}{x}\) and \(f_2(x)=\frac {9}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {9 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 {9}{x^{2}}\\ f_1 f_2 &=\frac {9}{x^{2}}\\ f_2^2 f_0 &=\frac {81}{x} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {9 u^{\prime \prime }\left (x \right )}{x}+\frac {81 u \left (x \right )}{x} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \sin \left (3 x \right )+c_{2} \cos \left (3 x \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = 3 c_{1} \cos \left (3 x \right )-3 c_{2} \sin \left (3 x \right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (3 c_{1} \cos \left (3 x \right )-3 c_{2} \sin \left (3 x \right )\right ) x}{9 \left (c_{1} \sin \left (3 x \right )+c_{2} \cos \left (3 x \right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\left (-c_{3} \cos \left (3 x \right )+\sin \left (3 x \right )\right ) x}{3 c_{3} \sin \left (3 x \right )+3 \cos \left (3 x \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (-c_{3} \cos \left (3 x \right )+\sin \left (3 x \right )\right ) x}{3 c_{3} \sin \left (3 x \right )+3 \cos \left (3 x \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (-c_{3} \cos \left (3 x \right )+\sin \left (3 x \right )\right ) x}{3 c_{3} \sin \left (3 x \right )+3 \cos \left (3 x \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x -y-9 y^{2}=x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y+x^{2}+9 y^{2}}{x} \end {array} \]
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: 0.0 (sec). Leaf size: 15
dsolve(x*diff(y(x),x)=y(x)+x^2+9*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\tan \left (3 x +3 c_{1} \right ) x}{3} \]
✓ Solution by Mathematica
Time used: 0.285 (sec). Leaf size: 17
DSolve[x*y'[x]==y[x]+x^2+9*y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{3} x \tan (3 (x+c_1)) \]