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$$\frac{1}{2}\frac{\partial^2w}{\partial y^2}+\sqrt{8x^2+25}\frac{\partial w}{\partial y}+4x^2w=0$$

"Find the general solution of the PDE in $w(x,y)$. State clearly any arbitrary functions that appear in general solutions."

Progression:

So far I tried using the method of separation, letting $$w(x,y) = X(x)Y(y)$$ after subbing the partial derivatives into the original PDE, I divided the equation by $X(x)Y(y)$ and I got the following:

$$\frac{1}{2}\frac{Y''(y)}{Y(y)} + \sqrt{8x^2+25}\frac{Y'(y)}{Y(y)}+4x^2=0 $$

so usually I would let $X$ on one side and $Y$ on the other side of the equation that = a constant $\gamma$ and decompose into 2 separate ODEs but in this case, I can't find a way to separate $X$ and $Y$.

Sorry for the previous post I will delete it. This is not an assignment just a past year's exam question I was practicing. Thanks in advance for any help!

Daniele Tampieri
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Jaymus Seah
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    Your differential equation only contains derivatives with respect to $y$. As such it's really just an ordinary DE in disguise, with coefficients that happen to depend on $x$. – Semiclassical Jan 26 '21 at 06:36
  • So I should work out the characteristics equation in terms of x? – Jaymus Seah Jan 26 '21 at 06:59
  • That's overkill: the idea is much simpler. See below for a hint. – Semiclassical Jan 26 '21 at 07:08
  • hmmm but how do I apply it to this question? Since it contains the second and first derivative in the same equation so I can't seem to integrate them directly. – Jaymus Seah Jan 26 '21 at 07:38
  • The lesson is not "integrate it like a linear DE." It's to solve it like you would a second-order ODE, and figure out how the x-dependence enters. – Semiclassical Jan 26 '21 at 14:45

1 Answers1

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Hint: Consider the simpler linear DE $\frac{\partial w}{\partial y}+x w=0$. This is equivalent to $$\frac{1}{w}\frac{\partial w}{\partial y}=\frac{\partial}{\partial y}\ln w=-x,$$

which we may immediately integrate w/r/t $y$ to obtain

$$\ln w(x,y)=-x y+C(x)\implies w(x,y)=A(x)e^{-x y}.$$

Note that this is basically equivalent to solving the linear ODE $\frac{dw}{dy}+xw=0$, treating $x$ as a constant, and then understanding that the coefficient of the general solution will depend on $x$ as well. Can you apply this insight to your DE?

Semiclassical
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