2.2.1 problem 1

Internal problem ID [18190]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 4. Autonomous systems. Exercises at page 69
Problem number : 1
Date solved : Thursday, December 19, 2024 at 06:17:48 PM
CAS classification : [_quadrature]

Solve

\begin{align*} x^{\prime }&=-\lambda x \end{align*}

Solved as first order autonomous ode

Time used: 0.148 (sec)

Integrating gives

\begin{align*} \int -\frac {1}{\lambda x}d x &= dt\\ -\frac {\ln \left (x \right )}{\lambda }&= t +c_1 \end{align*}

Singular solutions are found by solving

\begin{align*} -\lambda x&= 0 \end{align*}

for \(x\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} x = 0 \end{align*}

The following diagram is the phase line diagram. It classifies each of the above equilibrium points as stable or not stable or semi-stable.

Figure 2.46: Phase line diagram

Solving for \(x\) gives

\begin{align*} x &= 0 \\ x &= {\mathrm e}^{-c_1 \lambda -t \lambda } \\ \end{align*}

Summary of solutions found

\begin{align*} x &= 0 \\ x &= {\mathrm e}^{-c_1 \lambda -t \lambda } \\ \end{align*}

Solved as first order homogeneous class D2 ode

Time used: 0.121 (sec)

Applying change of variables \(x = u \left (t \right ) t\), then the ode becomes

\begin{align*} u^{\prime }\left (t \right ) t +u \left (t \right ) = -\lambda u \left (t \right ) t \end{align*}

Which is now solved The ode \(u^{\prime }\left (t \right ) = -\frac {u \left (t \right ) \left (t \lambda +1\right )}{t}\) is separable as it can be written as

\begin{align*} u^{\prime }\left (t \right )&= -\frac {u \left (t \right ) \left (t \lambda +1\right )}{t}\\ &= f(t) g(u) \end{align*}

Where

\begin{align*} f(t) &= -\frac {t \lambda +1}{t}\\ g(u) &= u \end{align*}

Integrating gives

\begin{align*} \int { \frac {1}{g(u)} \,du} &= \int { f(t) \,dt}\\ \int { \frac {1}{u}\,du} &= \int { -\frac {t \lambda +1}{t} \,dt}\\ \ln \left (u \left (t \right )\right )&=-t \lambda +\ln \left (\frac {1}{t}\right )+c_1 \end{align*}

We now need to find the singular solutions, these are found by finding for what values \(g(u)\) is zero, since we had to divide by this above. Solving \(g(u)=0\) or \(u=0\) for \(u \left (t \right )\) gives

\begin{align*} u \left (t \right )&=0 \end{align*}

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

Therefore the solutions found are

\begin{align*} \ln \left (u \left (t \right )\right ) = -t \lambda +\ln \left (\frac {1}{t}\right )+c_1\\ u \left (t \right ) = 0 \end{align*}

Solving for \(u \left (t \right )\) gives

\begin{align*} u \left (t \right ) &= 0 \\ u \left (t \right ) &= \frac {{\mathrm e}^{-t \lambda +c_1}}{t} \\ \end{align*}

Converting \(u \left (t \right ) = 0\) back to \(x\) gives

\begin{align*} x = 0 \end{align*}

Converting \(u \left (t \right ) = \frac {{\mathrm e}^{-t \lambda +c_1}}{t}\) back to \(x\) gives

\begin{align*} x = {\mathrm e}^{-t \lambda +c_1} \end{align*}

Summary of solutions found

\begin{align*} x &= 0 \\ x &= {\mathrm e}^{-t \lambda +c_1} \\ \end{align*}

Solved as first order Exact ode

Time used: 0.106 (sec)

To solve an ode of the form

\begin{equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A}\end{equation}

We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives

\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \]

Hence

\begin{equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B}\end{equation}

Comparing (A,B) shows that

\begin{align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end{align*}

But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that

\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\]

If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is

\[ M(t,x) \mathop {\mathrm {d}t}+ N(t,x) \mathop {\mathrm {d}x}=0 \tag {1A} \]

Therefore

\begin{align*} \mathop {\mathrm {d}x} &= \left (-\lambda x\right )\mathop {\mathrm {d}t}\\ \left (\lambda x\right ) \mathop {\mathrm {d}t} + \mathop {\mathrm {d}x} &= 0 \tag {2A} \end{align*}

Comparing (1A) and (2A) shows that

\begin{align*} M(t,x) &= \lambda x\\ N(t,x) &= 1 \end{align*}

The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied

\[ \frac {\partial M}{\partial x} = \frac {\partial N}{\partial t} \]

Using result found above gives

\begin{align*} \frac {\partial M}{\partial x} &= \frac {\partial }{\partial x} \left (\lambda x\right )\\ &= \lambda \end{align*}

And

\begin{align*} \frac {\partial N}{\partial t} &= \frac {\partial }{\partial t} \left (1\right )\\ &= 0 \end{align*}

Since \(\frac {\partial M}{\partial x} \neq \frac {\partial N}{\partial t}\), then the ODE is not exact. Since the ODE is not exact, we will try to find an integrating factor to make it exact. Let

\begin{align*} A &= \frac {1}{N} \left (\frac {\partial M}{\partial x} - \frac {\partial N}{\partial t} \right ) \\ &=1\left ( \left ( \lambda \right ) - \left (0 \right ) \right ) \\ &=\lambda \end{align*}

Since \(A\) does not depend on \(x\), then it can be used to find an integrating factor. The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{ \int A \mathop {\mathrm {d}t} } \\ &= e^{\int \lambda \mathop {\mathrm {d}t} } \end{align*}

The result of integrating gives

\begin{align*} \mu &= e^{t \lambda } \\ &= {\mathrm e}^{t \lambda } \end{align*}

\(M\) and \(N\) are multiplied by this integrating factor, giving new \(M\) and new \(N\) which are called \(\overline {M}\) and \(\overline {N}\) for now so not to confuse them with the original \(M\) and \(N\).

\begin{align*} \overline {M} &=\mu M \\ &= {\mathrm e}^{t \lambda }\left (\lambda x\right ) \\ &= \lambda x \,{\mathrm e}^{t \lambda } \end{align*}

And

\begin{align*} \overline {N} &=\mu N \\ &= {\mathrm e}^{t \lambda }\left (1\right ) \\ &= {\mathrm e}^{t \lambda } \end{align*}

Now a modified ODE is ontained from the original ODE, which is exact and can be solved. The modified ODE is

\begin{align*} \overline {M} + \overline {N} \frac { \mathop {\mathrm {d}x}}{\mathop {\mathrm {d}t}} &= 0 \\ \left (\lambda x \,{\mathrm e}^{t \lambda }\right ) + \left ({\mathrm e}^{t \lambda }\right ) \frac { \mathop {\mathrm {d}x}}{\mathop {\mathrm {d}t}} &= 0 \end{align*}

The following equations are now set up to solve for the function \(\phi \left (t,x\right )\)

\begin{align*} \frac {\partial \phi }{\partial t } &= \overline {M}\tag {1} \\ \frac {\partial \phi }{\partial x } &= \overline {N}\tag {2} \end{align*}

Integrating (2) w.r.t. \(x\) gives

\begin{align*} \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int \overline {N}\mathop {\mathrm {d}x} \\ \int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int {\mathrm e}^{t \lambda }\mathop {\mathrm {d}x} \\ \tag{3} \phi &= {\mathrm e}^{t \lambda } x+ f(t) \\ \end{align*}

Where \(f(t)\) is used for the constant of integration since \(\phi \) is a function of both \(t\) and \(x\). Taking derivative of equation (3) w.r.t \(t\) gives

\begin{equation} \tag{4} \frac {\partial \phi }{\partial t} = \lambda x \,{\mathrm e}^{t \lambda }+f'(t) \end{equation}

But equation (1) says that \(\frac {\partial \phi }{\partial t} = \lambda x \,{\mathrm e}^{t \lambda }\). Therefore equation (4) becomes

\begin{equation} \tag{5} \lambda x \,{\mathrm e}^{t \lambda } = \lambda x \,{\mathrm e}^{t \lambda }+f'(t) \end{equation}

Solving equation (5) for \( f'(t)\) gives

\[ f'(t) = 0 \]

Therefore

\[ f(t) = c_1 \]

Where \(c_1\) is constant of integration. Substituting this result for \(f(t)\) into equation (3) gives \(\phi \)

\[ \phi = {\mathrm e}^{t \lambda } x+ c_1 \]

But since \(\phi \) itself is a constant function, then let \(\phi =c_2\) where \(c_2\) is new constant and combining \(c_1\) and \(c_2\) constants into the constant \(c_1\) gives the solution as

\[ c_1 = {\mathrm e}^{t \lambda } x \]

Solving for \(x\) gives

\begin{align*} x &= c_1 \,{\mathrm e}^{-t \lambda } \\ \end{align*}

Summary of solutions found

\begin{align*} x &= c_1 \,{\mathrm e}^{-t \lambda } \\ \end{align*}

Solved using Lie symmetry for first order ode

Time used: 0.349 (sec)

Writing the ode as

\begin{align*} x^{\prime }&=-\lambda x\\ x^{\prime }&= \omega \left ( t,x\right ) \end{align*}

The condition of Lie symmetry is the linearized PDE given by

\begin{align*} \eta _{t}+\omega \left ( \eta _{x}-\xi _{t}\right ) -\omega ^{2}\xi _{x}-\omega _{t}\xi -\omega _{x}\eta =0\tag {A} \end{align*}

To determine \(\xi ,\eta \) then (A) is solved using ansatz. Making bivariate polynomials of degree 1 to use as anstaz gives

\begin{align*} \tag{1E} \xi &= t a_{2}+x a_{3}+a_{1} \\ \tag{2E} \eta &= t b_{2}+x b_{3}+b_{1} \\ \end{align*}

Where the unknown coefficients are

\[ \{a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\} \]

Substituting equations (1E,2E) and \(\omega \) into (A) gives

\begin{equation} \tag{5E} b_{2}-\lambda x \left (b_{3}-a_{2}\right )-\lambda ^{2} x^{2} a_{3}+\lambda \left (t b_{2}+x b_{3}+b_{1}\right ) = 0 \end{equation}

Putting the above in normal form gives

\[ -\lambda ^{2} x^{2} a_{3}+\lambda t b_{2}+\lambda x a_{2}+\lambda b_{1}+b_{2} = 0 \]

Setting the numerator to zero gives

\begin{equation} \tag{6E} -\lambda ^{2} x^{2} a_{3}+\lambda t b_{2}+\lambda x a_{2}+\lambda b_{1}+b_{2} = 0 \end{equation}

Looking at the above PDE shows the following are all the terms with \(\{t, x\}\) in them.

\[ \{t, x\} \]

The following substitution is now made to be able to collect on all terms with \(\{t, x\}\) in them

\[ \{t = v_{1}, x = v_{2}\} \]

The above PDE (6E) now becomes

\begin{equation} \tag{7E} -\lambda ^{2} a_{3} v_{2}^{2}+\lambda a_{2} v_{2}+\lambda b_{2} v_{1}+\lambda b_{1}+b_{2} = 0 \end{equation}

Collecting the above on the terms \(v_i\) introduced, and these are

\[ \{v_{1}, v_{2}\} \]

Equation (7E) now becomes

\begin{equation} \tag{8E} -\lambda ^{2} a_{3} v_{2}^{2}+\lambda a_{2} v_{2}+\lambda b_{2} v_{1}+\lambda b_{1}+b_{2} = 0 \end{equation}

Setting each coefficients in (8E) to zero gives the following equations to solve

\begin{align*} \lambda a_{2}&=0\\ \lambda b_{2}&=0\\ -\lambda ^{2} a_{3}&=0\\ \lambda b_{1}+b_{2}&=0 \end{align*}

Solving the above equations for the unknowns gives

\begin{align*} a_{1}&=a_{1}\\ a_{2}&=0\\ a_{3}&=0\\ b_{1}&=0\\ b_{2}&=0\\ b_{3}&=b_{3} \end{align*}

Substituting the above solution in the anstaz (1E,2E) (using \(1\) as arbitrary value for any unknown in the RHS) gives

\begin{align*} \xi &= 0 \\ \eta &= x \\ \end{align*}

The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( t,x\right ) \to \left ( R,S \right )\) 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 t}{\xi } &= \frac {d x}{\eta } = dS \tag {1} \end{align*}

The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial t} + \eta \frac {\partial }{\partial x}\right ) S(t,x) = 1\). 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 = t \end{align*}

\(S\) is found from

\begin{align*} S &= \int { \frac {1}{\eta }} dy\\ &= \int { \frac {1}{x}} dy \end{align*}

Which results in

\begin{align*} S&= \ln \left (x \right ) \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_{t} + \omega (t,x) S_{x} }{ R_{t} + \omega (t,x) R_{x} }\tag {2} \end{align*}

Where in the above \(R_{t},R_{x},S_{t},S_{x}\) are all partial derivatives and \(\omega (t,x)\) is the right hand side of the original ode given by

\begin{align*} \omega (t,x) &= -\lambda x \end{align*}

Evaluating all the partial derivatives gives

\begin{align*} R_{t} &= 1\\ R_{x} &= 0\\ S_{t} &= 0\\ S_{x} &= \frac {1}{x} \end{align*}

Substituting all the above in (2) and simplifying gives the ode in canonical coordinates.

\begin{align*} \frac {dS}{dR} &= -\lambda \tag {2A} \end{align*}

We now need to express the RHS as function of \(R\) only. This is done by solving for \(t,x\) in terms of \(R,S\) from the result obtained earlier and simplifying. This gives

\begin{align*} \frac {dS}{dR} &= -\lambda \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 \(R,S\).

Since the ode has the form \(\frac {d}{d R}S \left (R \right )=f(R)\), then we only need to integrate \(f(R)\).

\begin{align*} \int {dS} &= \int {-\lambda \, dR}\\ S \left (R \right ) &= -\lambda R + c_2 \end{align*}

To complete the solution, we just need to transform the above back to \(t,x\) coordinates. This results in

\begin{align*} \ln \left (x\right ) = -t \lambda +c_2 \end{align*}

Which gives

\begin{align*} x = {\mathrm e}^{-t \lambda +c_2} \end{align*}

Summary of solutions found

\begin{align*} x &= {\mathrm e}^{-t \lambda +c_2} \\ \end{align*}

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }=-\lambda x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=-\lambda x \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{x}=-\lambda \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{x}d t =\int -\lambda d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (x\right )=-t \lambda +\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x={\mathrm e}^{-t \lambda +\mathit {C1}} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful`
 

Maple dsolve solution

Solving time : 0.002 (sec)
Leaf size : 11

dsolve(diff(x(t),t) = -lambda*x(t), 
       x(t),singsol=all)
 

\[ x = c_1 \,{\mathrm e}^{-t \lambda } \]

Mathematica DSolve solution

Solving time : 0.021 (sec)
Leaf size : 18

DSolve[{D[x[t],t]==\[Lambda]*x[t],{}}, 
       x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 e^{\lambda t} \\ x(t)\to 0 \\ \end{align*}