Diff EQ help...

[ThundercrackeR]

Hello fellow mathematicians,

I started preparing my math exam (areas of differential equations, systems of Diff EQ, Laplace transformations etc.), and I am having problems with this particular equation (it comes from one of earlier exam terms). I tried various different ways to solve it, but somehow I always get stuck...

. . . . .\(\displaystyle \left(x^3\, +\, \sin(y)\right)\, \cdot\, y'\, =\, x^2\)

I tried replacing y' with (dy/dx) and then playing with that form... Separation of variables doesn't work... Tried to solve it as the homogenous diff EQ, but it seems that I can't get right integral to solve the equation. I guess that this (siny) gives me a lot of "trouble".

If I do get somewhere by myself, I will post the steps here... But in the meantime, please help! :grin:

 

[HallsofIvy]

My first thought was to write the equation as \(\displaystyle (x^3+ sin(y))dy= x^2dx\) and then as \(\displaystyle (x^3+ sin(y))dy- x^2dx= 0\). Now, a first order differential equation of the form \(\displaystyle f(x,y)dy+ g(x,y)dx= 0\) is "exact", and so easy to solve, if and only if \(\displaystyle f_x= g_y\). Here those are \(\displaystyle f(x,y)= x^3+ sin(y)\) and \(\displaystyle g(x,y)= x^2\) so that \(\displaystyle f_x= 3x^2\) and \(\displaystyle g_y= 0\). Those are NOT the same so the equation is not exact.

But the fact that \(\displaystyle g_y= 0\) suggests multiplying the equation by some function of y only so that \(\displaystyle g_y\) is not 0. If we multiply the equation by u(y) we have \(\displaystyle (x^3u(y)+ u(y)sin(y))dy- x^2u(y)dx= 0\).

Now, \(\displaystyle (x^3u(y)+ u(y)sin(y))_x= 3x^2u(u)\) and \(\displaystyle (-x^2u(y))_y= -x^2u'(y)\). In order that the equation now be exact we must have \(\displaystyle -x^2u'= 3x^2u\) which reduces to \(\displaystyle u'= -3u\), independent of x!