23)[ordinary differential equations; topic] Please solve the following problem Provide a well explained and understandable(readable) Step by step solution solution dy .y' - 2x?y = 0, y(0) = 5. Note: y = y(x) and y' = Y., хр%3DIn the power series solution of the above IVP find the coefficient of the X term to the nearestthousandth (3 decimal places). Do not calculate the entire power series.in the answerIn the answer box only put the coefficient part of the X term. Do not include the Xbox.

Given the differential equation, y'-2x2y=0 ⋯⋯(1) With the initial condition y(0)=5 Let…

Answered: dy y' - 2x?y = 0, y(0) = 5. Note: y =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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23)[ordinary differential equations; topic] Please solve the following problem Provide a well explained and understandable(readable) Step by step solution solution

dy .
y' - 2x?y = 0, y(0) = 5. Note: y = y(x) and y' = Y.
, хр
%3D
In the power series solution of the above IVP find the coefficient of the X term to the nearest
thousandth (3 decimal places). Do not calculate the entire power series.
in the answer
In the answer box only put the coefficient part of the X term. Do not include the X
box.

Expert Solution

Step 1

Given the differential equation,

$y' - 2 x^{2} y = 0 \dots \dots (1)$

With the initial condition $y (0) = 5$

Let $y (x) = \sum_{n = 0}^{\infty} a_{n} x^{n}$ be the solution of the differential equation (1).

Then,

$y' (x) = \sum_{n = 1}^{\infty} n a_{n} x^{n - 1}$

Here,

$\begin{array}{rcl} y (x) & = & a_{0} + a_{1} x + a_{2} x^{2} + \dots \\ y (0) & = & a_{0} \\ a_{0} & = & 5 \end{array}$

Step 2

Substituting in equation (1),

$\begin{array}{rcl} \sum_{n = 1}^{\infty} n a_{n} x^{n - 1} - 2 x^{2} \sum_{n = 0}^{\infty} a_{n} x^{n} & = & 0 \\ \sum_{n = 0}^{\infty} (n + 1) a_{n + 1} x^{n} - {\sum 2}_{n = 2}^{\infty} a_{n - 2} x^{n} & = & 0 \\ a_{1} + 2 a_{2} x + \sum_{n = 2}^{\infty} (n + 1) a_{n + 1} x^{n} - {\sum 2}_{n = 2}^{\infty} a_{n - 2} x^{n} & = & 0 \\ a_{1} + 2 a_{2} x + \sum_{n = 2}^{\infty} ((n + 1) a_{n + 1} - 2 a_{n - 2}) x^{n} & = & 0 \dots \dots (2) \end{array}$

Multiplying equation (2) by $x,$

$\begin{array}{rcl} a \end{array} \begin{array}{rcl} \end{array}$ ₁x+2a₂x²+∑n=2∞n+1an+1-2an-2xⁿ⁺¹=0 ⋯⋯(3)

Comparing the LHS with RHS of the equation (3),

$\begin{array}{rcl} a_{1} & = & 0 \\ a_{2} & = & 0 \\ (n + 1) a_{n + 1} - 2 a_{n - 2} & = & 0 \end{array}$

Then,

$a_{n + 1} = \frac{2 a_{n - 2}}{n + 1} \dots \dots (4)$

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