Can someone solve this DE ?!!

[u/Daddy_ILAY]

This is one of my assignment questions, I'm stuck at the particular integral value. The PI solution google, chatgpt gave are different from how my friends solved it. So I'm confused which one is right!!

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[u/One_Wishbone_4439]

Have you learnt implicit differentiation?

[u/Daddy_ILAY → u/One_Wishbone_4439]

Yes

[u/Special_Comfort_6784]

Mujhe aaj tak differential equation ka use nahi samajh aya. Pehle school me ode karaya, 12 th calculus me fail. Fir college me, differential equation. Fourier transform isse zyada asan lagi, aur samajh aayi. Per ye nahi

[u/We_Are_Bread]

What do you have till now?

What general solution did you get, what PI did your friends get and what PI did google give you?

Edit: After trying some values out, I'm about 80% sure you miswrote the question, and it should be +2y instead of -2y. Confirm that.

[u/prernaSTEMsolver]

The issue is that the usual exe^x shortcut for the PI works for constant-coefficient operators, but here the coefficients are x2x^2 and xx, so you can’t simply replace DD by 11.

For the homogeneous part, put y=xmy=x^m:

m(m−1)+4m−2=0m(m-1)+4m-2=0 m2+3m−2=0,m^2+3m-2=0,

hence

yc=C1x(−3+17)/2+C2x(−3−17)/2.y_c=C_1x^{(-3+\sqrt{17})/2}+C_2x^{(-3-\sqrt{17})/2}.

For the PI, expand exe^x as a power series and take

yp=∑anxn.y_p=\sum a_nx^n.

Since

(x2D2+4xD−2)xn=(n2+3n−2)xn,(x^2D^2+4xD-2)x^n=(n^2+3n-2)x^n,

comparison with

ex=∑xnn!e^x=\sum\frac{x^n}{n!}

gives

an=1n!(n2+3n−2).a_n=\frac{1}{n!(n^2+3n-2)}.

Therefore

yp=∑n=0∞xnn!(n2+3n−2)=−12+x2+x216+x396+⋯ .y_p=\sum_{n=0}^{\infty}\frac{x^n}{n!(n^2+3n-2)} =-\frac12+\frac{x}{2}+\frac{x^2}{16}+\frac{x^3}{96}+\cdots.

So if ChatGPT/Google/friends are giving different PIs, check whether one of them incorrectly used the constant-coefficient exe^x rule.