While solving second order non-homogenous differential equations of the form $y''+y'=5$, I realized that unlike while solving the ones of the form $y''+y'+y=Ax^n$ where we assume, $y_p=Ax^n+Bx^{n-1}+\cdots$, we assume in this case that $y_p=Ax$ instead of $A$. Can some one give me some insight. Or if this is wrong, how to find particular solution of this form (even though this is very basic).
Particular Solution of Differential Equations
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0For the particular solutions of linear ODEs, you don't have to be picky about how you get them. If it solves the equation, then it's fine. – 2012-12-11
2 Answers
You are looking for this particular solution because your right hand side has the form $ f(x)=5e^{0x}, $ and zero is a root of your characteristic polynomial $p(r)=r^2+r$. If it was not a root, then the form is $y_p=A$.
Added: The general rule as follows: if your right hand side has the form $ f(x)=P_n(x)e^{\alpha x}, $ where $P_n(x)$ is a polynomial of degree $n$ and $\alpha$ is a constant, then you should look for a particular solution in the form $ y_p(x)=Q_n(x)x^{k}e^{\alpha x}, $ where $Q_n(x)$ is a polynomial of degree $n$ with undetermined coefficients, $k$ is the multiplicity y of $\alpha$ as a root of characteristic polynomial (it can be also zero it is not a root).
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1no, you would assume generally $y_p=(Ax^2+Bx+C)x^2$ (however, how it should be clear from your particular problem $y_p=Ax^4$ will suffice). (see also my edit of the answer) – 2012-12-11
The reason is that any constant term will disappear when you differentiate it, so even if you put $y_p=Ax+B$, the term $B$ wouldn't appear at all when you wrote $y''_p + y'_p = 5$ in terms of $x$. Since you're looking for a particular solution, rather than a complete family of solutions, it's fine to drop it.
More generally, if you had $a_n\dfrac{d^ny}{dx^n} + a_{n-1} \dfrac{d^{n-1}y}{dx^{n-1}} + \cdots + a_{n-k}\dfrac{d^{k}y}{dx^{k}} = c$ (or even more general equations than this), where $0 \le k
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0Did you not mean $x^{n+k}$ as leading term. It contradicts with above observation. – 2012-12-11