Question

<e>Find a power series solution of 𝑥𝑦′=𝑦 .

Accepted Answer

By Power Series Method, the solution of the differential equation is y=c0∞∑n=0(x22)nn!=c0ex22, where c0 is any constant. Let us look at some details. Let y=∞∑n=0cnxn. By taking the derivative term by term, y=∞∑n=1ncnxn−1 Now, let us look at the differential equation. y=xy by substituting the above power series in the equation, ⇒∞∑n=1ncnxn−1=x⋅∞∑n=0cnxn by pulling the first term from the summation on the left, ⇒c1+∞∑n=2ncnxn−1=∞∑n=0cnxn+1 by shifting the indices of the summation on the left by 2,

Question #297043

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