Toronto Math Forum
MAT2442018F => MAT244Tests => Thanksgiving Bonus => Topic started by: Victor Ivrii on October 05, 2018, 05:49:09 PM

Lagrange equation is of the form
\begin{equation}
y= x\varphi (y')+\psi (y')
\label{eq1}
\end{equation}
with $\varphi(p)p\ne 0$. To solve it we plug $p=y'$ and differentiate equation:
\begin{equation}
pdx= \varphi(p)dx + \bigl(x\varphi'(p) +\psi'(p)\bigr)dp.
\label{eq2}
\end{equation}
This is a linear ODE with respect to $x$. We find the general solution $x=f(p,C)$ and then $y=f(p,c)\varphi(p)+\psi(p)$:
\begin{equation}
\left\{\begin{aligned}
&x=f(p,C)\\
&y=f(p,c)\varphi(p)+\psi(p)
\end{aligned}\right.
\label{eq4}
\end{equation}
gives us a solution in the parametric form.
(\ref{eq1}) can have a singular solution (or solutions)
\begin{equation}
y=x\varphi(c)+\psi(c),
\label{eq5}
\end{equation}
where $c$ is a root of equation $\varphi(c)c=0$.
Problem.
Find general and singular solutions to
$$y= 2xy'3(y')^2.$$

Find general and special solutions to $$y= 2xy'3(y')^2.$$
Substituting $y'=p$, we get an equation of the form $y=2xp–3{p^2}$
Differentiating both sides, we get
$dy=2xdp+2pdx6pdp$
$dy=pdx$ $\Rightarrow$ $pdx=2xdp+2pdx6pdp$ $\Rightarrow$ $–pdx=2xdp–6pdp$
Dividing by $p$, we get $dx={2x \over p}dp6dp$ $\Rightarrow$ $ {dx \over dp} + {2 \over p}x6=0 \tag{1}$
We get a linear differential equation. To solve $(1)$, we use an integrating factor given by $\mu=e^{\int{2 \over p}dp}=e^{2\lnp}=p^2$
Rewrite $(1)$ as ${2 \over p}x+{dx \over dp}=6$ and multiply by the integrating factor, $\mu(p)=p^2$ to get,
$$p^2{2 \over p}x+p^2{dx \over dp}=p^2 6$$
$${d \over dp}(p^2 x)=p^2 6$$
$$x={{2p^3+C} \over p^2}$$
Thus, the general solution to (1) is
$$x={{2p + {C \over p^2}}}\tag{2}$$
Substituting $(2)$ into the Lagrange Equation, we get
$$y={2\left({2p+\frac{C}{{{p^2}}}}\right)p–3p^2}=p^2+ \frac{2C}p$$
Thus,
\begin{equation} \left\{\begin{aligned} &x={{2p + {C \over p^2}}}\\ &y=p^2+ \frac{2C}p \end{aligned}\right. \tag{3}\end{equation}
gives us a solution in the parametric form.
The Lagrange equation, $y= 2xy'3(y')^2$, can also have a special solution (or solutions)
$\varphi (p)p=0$
$2p–p=0$ $\Rightarrow$ $p(2p)=0$ $\Rightarrow$ $p_1=0$ and $p_2=2$
Thus, the special solutions are
$$y=x\varphi (0)+\psi (0)=x \cdot 0+0=0.$$

Here is the solution

Monika, you are too greedy and wasted a good problem.
Zhuojing, I give you some points despite posting after Monika, but the next time such posts will be deleted in sight: in addition of being photo rather than the scan (http://forum.math.toronto.edu/index.php?topic=1078.0 (http://forum.math.toronto.edu/index.php?topic=1078.0)) which is the first sin, they are in the wrong orientation, so I was forced to chose between rotating the monitor ot myself (kidding) which is the insult added to the injury.