Internal problem ID [5259]
Internal file name [OUTPUT/4750_Friday_February_02_2024_05_11_16_AM_8396201/index.tex
]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 4. Equations of first order and first degree (Variable separable). Supplemetary
problems. Page 22
Problem number: 49.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati", "bernoulli", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_rational, _Bernoulli]
\[ \boxed {y^{2}+y x -y^{\prime } x=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(x,y)\\ &= \frac {y \left (x +y \right )}{x} \end {align*}
The \(x\) domain of \(f(x,y)\) when \(y=1\) is \[
\{x <0\boldsymbol {\lor }0 The \(x\) domain of \(\frac {\partial f}{\partial y}\) when \(y=1\) is \[
\{x <0\boldsymbol {\lor }0
Writing the ode as \begin {align*} y^{\prime }&=\frac {y \left (x +y \right )}{x}\\ y^{\prime }&= \omega \left ( x,y\right ) \end {align*}
The condition of Lie symmetry is the linearized PDE given by \begin {align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end {align*}
The type of this ode is known. It is of type ODE class Form \(\xi \) \(\eta \) linear ode \(y'=f(x) y(x) +g(x)\) \(0\) \(e^{\int fdx}\) separable ode \(y^{\prime }=f\left ( x\right ) g\left ( y\right ) \) \(\frac {1}{f}\) \(0\) quadrature ode \(y^{\prime }=f\left ( x\right ) \) \(0\) \(1\) quadrature ode \(y^{\prime }=g\left ( y\right ) \) \(1\) \(0\) homogeneous ODEs of
Class A \(y^{\prime }=f\left ( \frac {y}{x}\right ) \) \(x\) \(y\) homogeneous ODEs of
Class C \(y^{\prime }=\left ( a+bx+cy\right ) ^{\frac {n}{m}}\) \(1\) \(-\frac {b}{c}\) homogeneous class D \(y^{\prime }=\frac {y}{x}+g\left ( x\right ) F\left (\frac {y}{x}\right ) \) \(x^{2}\) \(xy\) First order special
form ID 1 \(y^{\prime }=g\left ( x\right ) e^{h\left (x\right ) +by}+f\left ( x\right ) \) \(\frac {e^{-\int bf\left ( x\right )dx-h\left ( x\right ) }}{g\left ( x\right ) }\) \(\frac {f\left ( x\right )e^{-\int bf\left ( x\right ) dx-h\left ( x\right ) }}{g\left ( x\right ) }\) polynomial type ode \(y^{\prime }=\frac {a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}\) \(\frac {a_{1}b_{2}x-a_{2}b_{1}x-b_{1}c_{2}+b_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}\) \(\frac {a_{1}b_{2}y-a_{2}b_{1}y-a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}\) Bernoulli ode \(y^{\prime }=f\left ( x\right ) y+g\left ( x\right ) y^{n}\) \(0\) \(e^{-\int \left ( n-1\right ) f\left ( x\right ) dx}y^{n}\) Reduced Riccati \(y^{\prime }=f_{1}\left ( x\right ) y+f_{2}\left ( x\right )y^{2}\) \(0\) \(e^{-\int f_{1}dx}\) The above table shows that \begin {align*} \xi \left (x,y\right ) &=0\\ \tag {A1} \eta \left (x,y\right ) &={\mathrm e}^{-x} y^{2} \end {align*}
The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\)
where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and
hence solved by integration.
The characteristic pde which is used to find the canonical coordinates is \begin {align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end {align*}
The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). Starting with the first pair of ode’s in (1)
gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Since
\(\xi =0\) then in this special case \begin {align*} R = x \end {align*}
\(S\) is found from \begin {align*} S &= \int { \frac {1}{\eta }} dy\\ &= \int { \frac {1}{{\mathrm e}^{-x} y^{2}}} dy \end {align*}
Which results in \begin {align*} S&= -\frac {{\mathrm e}^{x}}{y} \end {align*}
Now that \(R,S\) are found, we need to setup the ode in these coordinates. This is done by
evaluating \begin {align*} \frac {dS}{dR} &= \frac { S_{x} + \omega (x,y) S_{y} }{ R_{x} + \omega (x,y) R_{y} }\tag {2} \end {align*}
Where in the above \(R_{x},R_{y},S_{x},S_{y}\) are all partial derivatives and \(\omega (x,y)\) is the right hand side of the original ode
given by \begin {align*} \omega (x,y) &= \frac {y \left (x +y \right )}{x} \end {align*}
Evaluating all the partial derivatives gives \begin {align*} R_{x} &= 1\\ R_{y} &= 0\\ S_{x} &= -\frac {{\mathrm e}^{x}}{y}\\ S_{y} &= \frac {{\mathrm e}^{x}}{y^{2}} \end {align*}
Substituting all the above in (2) and simplifying gives the ode in canonical coordinates.
\begin {align*} \frac {dS}{dR} &= \frac {{\mathrm e}^{x}}{x}\tag {2A} \end {align*}
We now need to express the RHS as function of \(R\) only. This is done by solving for \(x,y\) in terms of
\(R,S\) from the result obtained earlier and simplifying. This gives \begin {align*} \frac {dS}{dR} &= \frac {{\mathrm e}^{R}}{R} \end {align*}
The above is a quadrature ode. This is the whole point of Lie symmetry method. It
converts an ode, no matter how complicated it is, to one that can be solved by
integration when the ode is in the canonical coordiates \(R,S\). Integrating the above gives
\begin {align*} S \left (R \right ) = -\operatorname {expIntegral}_{1}\left (-R \right )+c_{1}\tag {4} \end {align*}
To complete the solution, we just need to transform (4) back to \(x,y\) coordinates. This results in
\begin {align*} -\frac {{\mathrm e}^{x}}{y} = -\operatorname {expIntegral}_{1}\left (-x \right )+c_{1} \end {align*}
Which simplifies to \begin {align*} -\frac {{\mathrm e}^{x}}{y} = -\operatorname {expIntegral}_{1}\left (-x \right )+c_{1} \end {align*}
Which gives \begin {align*} y = \frac {{\mathrm e}^{x}}{\operatorname {expIntegral}_{1}\left (-x \right )-c_{1}} \end {align*}
The following diagram shows solution curves of the original ode and how they transform in
the canonical coordinates space using the mapping shown.
Original ode in \(x,y\) coordinates
Canonical
coordinates
transformation ODE in canonical coordinates \((R,S)\) \( \frac {dy}{dx} = \frac {y \left (x +y \right )}{x}\) \( \frac {d S}{d R} = \frac {{\mathrm e}^{R}}{R}\) \(\!\begin {aligned} R&= x\\ S&= -\frac {{\mathrm e}^{x}}{y} \end {aligned} \) Initial conditions are used to solve for \(c_{1}\). Substituting \(x=1\) and \(y=1\) in the above solution gives an
equation to solve for the constant of integration. \begin {align*} 1 = \frac {{\mathrm e}}{\operatorname {expIntegral}_{1}\left (-1\right )-c_{1}} \end {align*}
The solutions are \begin {align*} c_{1} = \operatorname {expIntegral}_{1}\left (-1\right )-{\mathrm e} \end {align*}
Trying the constant \begin {align*} c_{1} = \operatorname {expIntegral}_{1}\left (-1\right )-{\mathrm e} \end {align*}
Substituting this in the general solution gives \begin {align*} y&=-\frac {{\mathrm e}^{x}}{-\operatorname {expIntegral}_{1}\left (-x \right )+\operatorname {expIntegral}_{1}\left (-1\right )-{\mathrm e}} \end {align*}
The constant \(c_{1} = \operatorname {expIntegral}_{1}\left (-1\right )-{\mathrm e}\) gives valid solution.
Summary
The solution(s) found are the following \begin{align*}
\tag{1} y &= -\frac {{\mathrm e}^{x}}{-\operatorname {expIntegral}_{1}\left (-x \right )+\operatorname {expIntegral}_{1}\left (-1\right )-{\mathrm e}} \\
\end{align*} Verification of solutions
\[
y = -\frac {{\mathrm e}^{x}}{-\operatorname {expIntegral}_{1}\left (-x \right )+\operatorname {expIntegral}_{1}\left (-1\right )-{\mathrm e}}
\] Verified OK. In canonical form, the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y \left (x +y \right )}{x} \end {align*}
This is a Bernoulli ODE. \[ y' = y +\frac {1}{x} y^{2} \tag {1} \] The standard Bernoulli ODE has the form \[ y' = f_0(x)y+f_1(x)y^n \tag {2} \] The first step is to
divide the above equation by \(y^n \) which gives \[ \frac {y'}{y^n} = f_0(x) y^{1-n} +f_1(x) \tag {3} \] The next step is use the substitution \(w = y^{1-n}\) in equation
(3) which generates a new ODE in \(w \left (x \right )\) which will be linear and can be easily solved using an
integrating factor. Backsubstitution then gives the solution \(y(x)\) which is what we
want.
This method is now applied to the ODE at hand. Comparing the ODE (1) With (2) Shows
that \begin {align*} f_0(x)&=1\\ f_1(x)&=\frac {1}{x}\\ n &=2 \end {align*}
Dividing both sides of ODE (1) by \(y^n=y^{2}\) gives \begin {align*} y'\frac {1}{y^{2}} &= \frac {1}{y} +\frac {1}{x} \tag {4} \end {align*}
Let \begin {align*} w &= y^{1-n} \\ &= \frac {1}{y} \tag {5} \end {align*}
Taking derivative of equation (5) w.r.t \(x\) gives \begin {align*} w' &= -\frac {1}{y^{2}}y' \tag {6} \end {align*}
Substituting equations (5) and (6) into equation (4) gives \begin {align*} -w^{\prime }\left (x \right )&= w \left (x \right )+\frac {1}{x}\\ w' &= -w -\frac {1}{x} \tag {7} \end {align*}
The above now is a linear ODE in \(w \left (x \right )\) which is now solved.
Entering Linear first order ODE solver. In canonical form a linear first order is
\begin {align*} w^{\prime }\left (x \right ) + p(x)w \left (x \right ) &= q(x) \end {align*}
Where here \begin {align*} p(x) &=1\\ q(x) &=-\frac {1}{x} \end {align*}
Hence the ode is \begin {align*} w^{\prime }\left (x \right )+w \left (x \right ) = -\frac {1}{x} \end {align*}
The integrating factor \(\mu \) is \begin{align*}
\mu &= {\mathrm e}^{\int 1d x} \\
&= {\mathrm e}^{x} \\
\end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu w\right ) &= \left (\mu \right ) \left (-\frac {1}{x}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left ({\mathrm e}^{x} w\right ) &= \left ({\mathrm e}^{x}\right ) \left (-\frac {1}{x}\right )\\ \mathrm {d} \left ({\mathrm e}^{x} w\right ) &= \left (-\frac {{\mathrm e}^{x}}{x}\right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} {\mathrm e}^{x} w &= \int {-\frac {{\mathrm e}^{x}}{x}\,\mathrm {d} x}\\ {\mathrm e}^{x} w &= \operatorname {expIntegral}_{1}\left (-x \right ) + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{x}\) results in \begin {align*} w \left (x \right ) &= {\mathrm e}^{-x} \operatorname {expIntegral}_{1}\left (-x \right )+c_{1} {\mathrm e}^{-x} \end {align*}
which simplifies to \begin {align*} w \left (x \right ) &= {\mathrm e}^{-x} \left (\operatorname {expIntegral}_{1}\left (-x \right )+c_{1} \right ) \end {align*}
Replacing \(w\) in the above by \(\frac {1}{y}\) using equation (5) gives the final solution. \begin {align*} \frac {1}{y} = {\mathrm e}^{-x} \left (\operatorname {expIntegral}_{1}\left (-x \right )+c_{1} \right ) \end {align*}
Or \begin {align*} y = \frac {{\mathrm e}^{x}}{\operatorname {expIntegral}_{1}\left (-x \right )+c_{1}} \end {align*}
Which is simplified to \[ y = \frac {{\mathrm e}^{x}}{\operatorname {expIntegral}_{1}\left (-x \right )+c_{1}} \] Initial conditions are used to solve for \(c_{1}\). Substituting \(x=1\) and \(y=1\)
in the above solution gives an equation to solve for the constant of integration.
\begin {align*} 1 = \frac {{\mathrm e}}{\operatorname {expIntegral}_{1}\left (-1\right )+c_{1}} \end {align*}
The solutions are \begin {align*} c_{1} = -\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e} \end {align*}
Trying the constant \begin {align*} c_{1} = -\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e} \end {align*}
Substituting this in the general solution gives \begin {align*} y&=\frac {{\mathrm e}^{x}}{\operatorname {expIntegral}_{1}\left (-x \right )-\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e}} \end {align*}
The constant \(c_{1} = -\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e}\) gives valid solution.
Summary
The solution(s) found are the following \begin{align*}
\tag{1} y &= \frac {{\mathrm e}^{x}}{\operatorname {expIntegral}_{1}\left (-x \right )-\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e}} \\
\end{align*} Verification of solutions
\[
y = \frac {{\mathrm e}^{x}}{\operatorname {expIntegral}_{1}\left (-x \right )-\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e}}
\] Verified OK. In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y \left (x +y \right )}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y +\frac {y^{2}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows that \(f_0(x)=0\), \(f_1(x)=1\) and \(f_2(x)=\frac {1}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {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 {1}{x^{2}}\\ f_1 f_2 &=\frac {1}{x}\\ f_2^2 f_0 &=0 \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {u^{\prime \prime }\left (x \right )}{x}-\left (-\frac {1}{x^{2}}+\frac {1}{x}\right ) u^{\prime }\left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[
u \left (x \right ) = c_{1} +\operatorname {expIntegral}_{1}\left (-x \right ) c_{2}
\] The above shows that \[
u^{\prime }\left (x \right ) = -\frac {{\mathrm e}^{x} c_{2}}{x}
\] Using the above in (1) gives the solution \[
y = \frac {{\mathrm e}^{x} c_{2}}{c_{1} +\operatorname {expIntegral}_{1}\left (-x \right ) c_{2}}
\] Dividing both
numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following
solution
\[
y = \frac {{\mathrm e}^{x}}{c_{3} +\operatorname {expIntegral}_{1}\left (-x \right )}
\] Initial conditions are used to solve for \(c_{3}\). Substituting \(x=1\) and \(y=1\) in the above solution gives an
equation to solve for the constant of integration. \begin {align*} 1 = \frac {{\mathrm e}}{c_{3} +\operatorname {expIntegral}_{1}\left (-1\right )} \end {align*}
The solutions are \begin {align*} c_{3} = -\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e} \end {align*}
Trying the constant \begin {align*} c_{3} = -\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e} \end {align*}
Substituting this in the general solution gives \begin {align*} y&=\frac {{\mathrm e}^{x}}{\operatorname {expIntegral}_{1}\left (-x \right )-\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e}} \end {align*}
The constant \(c_{3} = -\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e}\) gives valid solution.
Summary
The solution(s) found are the following \begin{align*}
\tag{1} y &= \frac {{\mathrm e}^{x}}{\operatorname {expIntegral}_{1}\left (-x \right )-\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e}} \\
\end{align*} Verification of solutions
\[
y = \frac {{\mathrm e}^{x}}{\operatorname {expIntegral}_{1}\left (-x \right )-\operatorname {expIntegral}_{1}\left (-1\right )+{\mathrm e}}
\] Verified OK. \[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{2}+y x -y^{\prime } x =0, y \left (1\right )=1\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2}+y x}{x} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (1\right )=1 \\ {} & {} & 0 \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} 0 \\ {} & {} & 0=0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} 0=0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & 0 \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & 0 \end {array} \]
Maple trace
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 22
\[
y \left (x \right ) = \frac {{\mathrm e}^{x}}{\operatorname {expIntegral}_{1}\left (-x \right )+{\mathrm e}-\operatorname {expIntegral}_{1}\left (-1\right )}
\]
✓ Solution by Mathematica
Time used: 0.161 (sec). Leaf size: 19
\[
y(x)\to \frac {e^x}{-\operatorname {ExpIntegralEi}(x)+\operatorname {ExpIntegralEi}(1)+e}
\]
2.24.2 Solving as first order ode lie symmetry lookup ode
Bernoulli
. Therefore we do not need
to solve the PDE (A), and can just use the lookup table shown below to find \(\xi ,\eta \)
2.24.3 Solving as bernoulli ode
2.24.4 Solving as riccati ode
2.24.5 Maple step by step solution
`Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
<- Bernoulli successful`
dsolve([(y(x)^2+x*y(x))-x*diff(y(x),x)= 0,y(1) = 1],y(x), singsol=all)
DSolve[{(y[x]^2+x*y[x])-x*y'[x]== 0,{y[1]==1}},y[x],x,IncludeSingularSolutions -> True]