4. Find an integrating factor \(\displaystyle \mu(y),\) depending only on y, for the differential equation:
. . . . .\(\displaystyle 2y^2\, (x\, +\, y^2)\, +\, xy\, (x\, +\, 6y^2)\, \dfrac{dy}{dx}\, =\, 0\)
...and hence find the general solution of this equation.
5. Find a polynomial \(\displaystyle p(x)\) so that \(\displaystyle p(x)\, e^{2x}\) is a solution of the second-order differential equation:
. . . . .\(\displaystyle \dfrac{d^2y}{dx^2}\, -\, 4\, \dfrac{dy}{dx}\, +\, 4y\, =\, x^2\, e^{2x}\)
...with the conditions that \(\displaystyle y(0)\, =\, 1\) and \(\displaystyle y(1)\, =\, 0.\)
I need help with both questions. For the 4 I'm not sure how to start it, I've looked up how to do those questions but can't understand the formatting.
For 5 I set y=p(x)e^2x and differentiated to the point where I eliminated p(x) and p'(x) but am not sure where to go from there. Any help is appreciated
Thanks
. . . . .\(\displaystyle 2y^2\, (x\, +\, y^2)\, +\, xy\, (x\, +\, 6y^2)\, \dfrac{dy}{dx}\, =\, 0\)
...and hence find the general solution of this equation.
5. Find a polynomial \(\displaystyle p(x)\) so that \(\displaystyle p(x)\, e^{2x}\) is a solution of the second-order differential equation:
. . . . .\(\displaystyle \dfrac{d^2y}{dx^2}\, -\, 4\, \dfrac{dy}{dx}\, +\, 4y\, =\, x^2\, e^{2x}\)
...with the conditions that \(\displaystyle y(0)\, =\, 1\) and \(\displaystyle y(1)\, =\, 0.\)
I need help with both questions. For the 4 I'm not sure how to start it, I've looked up how to do those questions but can't understand the formatting.
For 5 I set y=p(x)e^2x and differentiated to the point where I eliminated p(x) and p'(x) but am not sure where to go from there. Any help is appreciated
Thanks
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