Question

solve dy/dx = 1-2xy given that x(0)=0 by taylor sixost method​

Answer 1

Answer:

dy dx. = f (x,y) given y(x0) = y0. (1) to study the various numerical ... The methods of Picard and Taylor series belong to the former class of ... φ(x, y, c1, c2, ..., cn)=0. ... 5. Find the solution of dy dx. =1+ xy which passes through (0,1) in the ... y(0.1) correct to 3 decimal places from dy dx. + 2xy = 1 with y(0) = 0

Step-by-step explanation:

dy dx. = f (x,y) given y(x0) = y0. (1) to study the various numerical ... The methods of Picard and Taylor series belong to the former class of ... φ(x, y, c1, c2, ..., cn)=0. ... 5. Find the solution of dy dx. =1+ xy which passes through (0,1) in the ... y(0.1) correct to 3 decimal places from dy dx. + 2xy = 1 with y(0) = 0

Answer 2

Answer: y(x) = $\frac{x}{2}$ - $\frac{x^{4} }{8}$ + O($x^{6}$)

Note that the O($x^{6}$) term represents the error in our approximation        and gets smaller as we include more terms in the Taylor series expansion.

Given:

         $\frac{dy}{dx}$ = 1-2xy

         x(0) = 0

To Find:

          Solution of $\frac{dy}{dx}$ = 1 - 2xy such that x(0) = 0

Solution:

To solve the given differential equation using the Taylor series method,           we have to first find the first few terms of the Taylor series expansion of the solution y(x) around x=0.

                Let y(x) = $a_{0}$ + $a_{1}$x + $a_{2}$$x^{2}$ + $a_{3}$$x^{3}$ + $a_{4}$$x^{4}$ + $a_{5}$$x^{5}$ + O($x^{6}$)

           

        Differentiating y(x), we get:

                         $\frac{dy}{dx}$ = $a_{1}$ + 2$a_{2}$x + 3$a_{3}$$x^{2}$ + 4$a_{4}$$x^{3}$ + 5$a_{5}$$x^{4}$ + O($x^{5}$)

Substituting these into the given differential equation and equating coefficients of like powers of x, we get:

                    a1 = 0 (since x(0) = 0)

                    a2 = 1/2 (from the coefficient of x in the given equation)

                    a3 = 0 (from the coefficient of $x^{2}$ in the given equation)

                    a4 = -1/8 (from the coefficient of $x^{3}$ in the given equation)

                    a5 = 0 (from the coefficient of $x^{4}$ in the given equation)

Therefore, the solution to the given differential equation using the Taylor series method up to sixth order is:

                                y(x) = $\frac{x}{2}$ - $\frac{x^{4} }{8}$ + O($x^{6}$)

where,

      O($x^{6}$) term represents the error in our approximation and gets   smaller as we include more terms in the Taylor series expansion.

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