D'Alembert equation

The D'Alembert equation is a differential equation of the form

y=x\varphi (y')+f(y'),

{\ displaystyle y = x \ varphi (y ') + f (y'),}

{\ displaystyle y = x \ varphi (y ') + f (y'),}

Where $\varphi$ ${\ displaystyle \ varphi}$ $\ varphi$ and $f$ ${\ displaystyle f}$ $f$ - functions. It was first studied by J. D'Alembert (J. D'Alembert, 1748). Also known as the Lagrange equations , a special case for $\varphi (y')\equiv y'$ ${\ displaystyle \ varphi (y ') \ equiv y'}$ ${\ displaystyle \ varphi (y ') \ equiv y'}$ called the Claireau equation ^[1] .

Content

Solution

The integration of differential equations of this type is carried out in a parametric form , using the parameter

y'=p.

{\ displaystyle y '= p.}

Given this substitution, the original equation takes the form

y=x\varphi (p)+f(p).

{\ displaystyle y = x \ varphi (p) + f (p).}

Differentiation by x gives:

p=\varphi (p)+\left(x\varphi '(p)+f'(p)\right){\frac {dp}{dx}}

{\ displaystyle p = \ varphi (p) + \ left (x \ varphi '(p) + f' (p) \ right) {\ frac {dp} {dx}}}

or

p-\varphi (p)=\left(x\varphi '(p)+f'(p)\right){\frac {dp}{dx}}.

{\ displaystyle p- \ varphi (p) = \ left (x \ varphi '(p) + f' (p) \ right) {\ frac {dp} {dx}}.}

Special Solutions

One solution to the last equation is any function whose derivative is constant $y'=p=p_{0}$ ${\ displaystyle y '= p = p_ {0}}$ satisfying the algebraic equation

p_{0}-\varphi (p_{0})=0,

{\ displaystyle p_ {0} - \ varphi (p_ {0}) = 0,}

since for constant $p_{0}$ ${\ displaystyle p_ {0}}$

{\frac {dp}{dx}}\equiv 0.

{\ displaystyle {\ frac {dp} {dx}} \ equiv 0.}

If a $y'=p_{0}$ ${\ displaystyle y '= p_ {0}}$ then $y=p_{0}x+C_{0}$ ${\ displaystyle y = p_ {0} x + C_ {0}}$ , the constant C should be found by substitution in the original equation:

p_{0}x+C_{0}=x\varphi (p_{0})+f(p_{0}),

{\ displaystyle p_ {0} x + C_ {0} = x \ varphi (p_ {0}) + f (p_ {0}),}

since in this case $p_{0}=\varphi (p_{0})$ ${\ displaystyle p_ {0} = \ varphi (p_ {0})}$ then

C_{0}=f(p_{0})

{\ displaystyle C_ {0} = f (p_ {0})}

.

Finally, we can write:

y=x\varphi (p_{0})+f(p_{0})

{\ displaystyle y = x \ varphi (p_ {0}) + f (p_ {0})}

.

If such a solution cannot be obtained from the general, then it is called special .

General decision

We consider the inverse function k $p=y'$ ${\ displaystyle p = y '}$ , then, using the theorem on the derivative of the inverse function, we can write:

{\frac {dx}{dp}}-x{\frac {\varphi (p)}{p-\varphi (p)}}={\frac {f(p)}{p-\varphi (p)}}

{\ displaystyle {\ frac {dx} {dp}} - x {\ frac {\ varphi (p)} {p- \ varphi (p)}} = {\ frac {f (p)} {p- \ varphi (p)}}}

.

This equation is a linear differential equation of the first order , solving which, we obtain the expression for x as a function of p :

x=w(p,C).

{\ displaystyle x = w (p, C).}

Thus, we obtain a solution to the initial differential equation in a parametric form:

{\begin{cases}y=x\varphi (p)+f(p)\\x=w(p,C)\end{cases}}

{\ displaystyle {\ begin {cases} y = x \ varphi (p) + f (p) \\ x = w (p, C) \ end {cases}}}

.

Eliminating the variable p from this system, we obtain the general solution in the form

\Phi (x,y,C)=0

{\ displaystyle \ Phi (x, y, C) = 0}

.

Notes

↑ Piskunov H. S. Differential and integral calculus for technical colleges, vol. 2 .: Textbook for technical colleges .. - 13th ed .. - M: Nauka, Main Edition of Physics and Mathematics, 1985. - P. 46- 48. - 560 s.

[1] Piskunov H. S. Differential and integral calculus for technical colleges, vol. 2 .: Textbook for technical colleges .. - 13th ed .. - M: Nauka, Main Edition of Physics and Mathematics, 1985. - P. 46- 48. - 560 s.