91Ó°ÊÓ

Given the equation \[M dx + N dy = 0,\] and the transformation \[x=k \zeta,\quad y=k \eta,\] we want to show that the equation remains invariant under this transformation. To show this, we differentiate both \(x\) and \(y\) with respect to \(\zeta,\) we get $$ \frac{dx}{d\zeta} = k, \qquad \frac{dy}{d\eta}=k $$ Now, we substitute these into the given equation, we get $$ M(k d\zeta) + N(k d\eta) = 0 $$ $$ k(M d\zeta + N d\eta) = 0 $$ Since \(k\) is a non-zero constant, the equation above holds if \(M d\zeta + N d\eta = 0\). Thus, the equation is invariant under the transformation.

Now we want to show that the general solution of Equation (A) can be written in the form $$ x = c \phi\left(\frac{y}{x}\right) $$ To do this, we can use the substitution \(v = \frac{y}{x}\) or \(y= xv\). Now take the derivative of \(y\) with respect to \(x\), we get $$ \frac{dy}{dx} = x\frac{dv}{dx} + v $$ Substitute the above expression and \(y = xv\) into the given equation: $$ M dx + N(x\frac{dv}{dx} + v) dx = 0 $$ This can be simplified as, $$ M + N(x\frac{dv}{dx} + v) = 0 $$ Now, rearrange the equation to get a separated equation: $$ \frac{dv}{dx} = - \frac{M + Nv}{Nx} $$ Integrate both sides to obtain the general solution: $$ \int \frac{dv}{M + Nv} = - \int \frac{dx}{x} $$ Let \(\phi(v) = \int \frac{dv}{M + Nv}\). Integrating both sides we have: $$ \phi(v) = -\ln |x| + \ln|c| $$ Exponentiating both sides and solving for x we get $$ x = c \phi\left(\frac{y}{x}\right) $$ which proves the result.

Given the transformation: $$ x=k \zeta, \quad y=k \eta $$ and the general solution of Equation (A), which is: $$ x = c \phi\left(\frac{y}{x}\right) $$ We want to show that the solution is invariant under the transformation. Substituting the transformation into the general solution, we get $$ k \zeta = c \phi\left(\frac{k \eta}{k \zeta}\right) $$ Divide both sides by \(k\): $$ \zeta = \frac{c}{k} \phi\left(\frac{\eta}{\zeta}\right) $$ Now, we can see that the solution has the same form as the given general solution. Thus, the solution is invariant under the transformation.

The geometric interpretation of the results proved in (a) and (c) can be summarized as follows: 1. Invariance under the transformation in (a) signifies that any solution to the given homogeneous differential equation remains invariant under the transformation \(x=k\zeta, \quad y=k\eta\). In simpler terms, it means that the graph of the solution is scale-invariant. The family of solutions will continue to maintain its geometric feature and properties when scaled by a constant factor k. 2. Invariance under the transformation in (c) refers to the invariant properties of the general solution of the given homogeneous differential equation. This means that the transformation preserves the underlying structure of the solution space, showing the self-similarity nature of the family of solutions under the transformation.