Solve y=x+frac{x^{3}}{3}+frac{x^{5}}{5}+x^7/7+x^9/9+x^11/11+x^13/13

$$y = x + \frac { x ^ { 3 } } { 3 } + \frac { x ^ { 5 } } { 5 } + \frac { x ^ { 7 } } { 7 } + \frac { x ^ { 9 } } { 9 } + \frac { x ^ { 11 } } { 11 } + \frac { x ^ { 13 } } { 13 }$$

Solve for y

$$y=\frac{x^{13}}{13}+\frac{x^{11}}{11}+\frac{x^{9}}{9}+\frac{x^{7}}{7}+\frac{x^{5}}{5}+\frac{x^{3}}{3}+x$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 5 is 15. Multiply \frac{x^{3}}{3} times \frac{5}{5}. Multiply \frac{x^{5}}{5} times \frac{3}{3}.

$$y=x+\frac{5x^{3}}{15}+\frac{3x^{5}}{15}+\frac{x^{7}}{7}+\frac{x^{9}}{9}+\frac{x^{11}}{11}+\frac{x^{13}}{13}$$

Since \frac{5x^{3}}{15} and \frac{3x^{5}}{15} have the same denominator, add them by adding their numerators.

$$y=x+\frac{5x^{3}+3x^{5}}{15}+\frac{x^{7}}{7}+\frac{x^{9}}{9}+\frac{x^{11}}{11}+\frac{x^{13}}{13}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 15 and 7 is 105. Multiply \frac{5x^{3}+3x^{5}}{15} times \frac{7}{7}. Multiply \frac{x^{7}}{7} times \frac{15}{15}.

$$y=x+\frac{7\left(5x^{3}+3x^{5}\right)}{105}+\frac{15x^{7}}{105}+\frac{x^{9}}{9}+\frac{x^{11}}{11}+\frac{x^{13}}{13}$$

Since \frac{7\left(5x^{3}+3x^{5}\right)}{105} and \frac{15x^{7}}{105} have the same denominator, add them by adding their numerators.

$$y=x+\frac{7\left(5x^{3}+3x^{5}\right)+15x^{7}}{105}+\frac{x^{9}}{9}+\frac{x^{11}}{11}+\frac{x^{13}}{13}$$

Do the multiplications in 7\left(5x^{3}+3x^{5}\right)+15x^{7}.

$$y=x+\frac{35x^{3}+21x^{5}+15x^{7}}{105}+\frac{x^{9}}{9}+\frac{x^{11}}{11}+\frac{x^{13}}{13}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 105 and 9 is 315. Multiply \frac{35x^{3}+21x^{5}+15x^{7}}{105} times \frac{3}{3}. Multiply \frac{x^{9}}{9} times \frac{35}{35}.

$$y=x+\frac{3\left(35x^{3}+21x^{5}+15x^{7}\right)}{315}+\frac{35x^{9}}{315}+\frac{x^{11}}{11}+\frac{x^{13}}{13}$$

Since \frac{3\left(35x^{3}+21x^{5}+15x^{7}\right)}{315} and \frac{35x^{9}}{315} have the same denominator, add them by adding their numerators.

$$y=x+\frac{3\left(35x^{3}+21x^{5}+15x^{7}\right)+35x^{9}}{315}+\frac{x^{11}}{11}+\frac{x^{13}}{13}$$

Do the multiplications in 3\left(35x^{3}+21x^{5}+15x^{7}\right)+35x^{9}.

$$y=x+\frac{105x^{3}+63x^{5}+45x^{7}+35x^{9}}{315}+\frac{x^{11}}{11}+\frac{x^{13}}{13}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 315 and 11 is 3465. Multiply \frac{105x^{3}+63x^{5}+45x^{7}+35x^{9}}{315} times \frac{11}{11}. Multiply \frac{x^{11}}{11} times \frac{315}{315}.

$$y=x+\frac{11\left(105x^{3}+63x^{5}+45x^{7}+35x^{9}\right)}{3465}+\frac{315x^{11}}{3465}+\frac{x^{13}}{13}$$

Since \frac{11\left(105x^{3}+63x^{5}+45x^{7}+35x^{9}\right)}{3465} and \frac{315x^{11}}{3465} have the same denominator, add them by adding their numerators.

$$y=x+\frac{11\left(105x^{3}+63x^{5}+45x^{7}+35x^{9}\right)+315x^{11}}{3465}+\frac{x^{13}}{13}$$

Do the multiplications in 11\left(105x^{3}+63x^{5}+45x^{7}+35x^{9}\right)+315x^{11}.

$$y=x+\frac{1155x^{3}+693x^{5}+495x^{7}+385x^{9}+315x^{11}}{3465}+\frac{x^{13}}{13}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3465 and 13 is 45045. Multiply \frac{1155x^{3}+693x^{5}+495x^{7}+385x^{9}+315x^{11}}{3465} times \frac{13}{13}. Multiply \frac{x^{13}}{13} times \frac{3465}{3465}.

$$y=x+\frac{13\left(1155x^{3}+693x^{5}+495x^{7}+385x^{9}+315x^{11}\right)}{45045}+\frac{3465x^{13}}{45045}$$

Since \frac{13\left(1155x^{3}+693x^{5}+495x^{7}+385x^{9}+315x^{11}\right)}{45045} and \frac{3465x^{13}}{45045} have the same denominator, add them by adding their numerators.

$$y=x+\frac{13\left(1155x^{3}+693x^{5}+495x^{7}+385x^{9}+315x^{11}\right)+3465x^{13}}{45045}$$

Do the multiplications in 13\left(1155x^{3}+693x^{5}+495x^{7}+385x^{9}+315x^{11}\right)+3465x^{13}.

$$y=x+\frac{15015x^{3}+9009x^{5}+6435x^{7}+5005x^{9}+4095x^{11}+3465x^{13}}{45045}$$

Divide each term of 15015x^{3}+9009x^{5}+6435x^{7}+5005x^{9}+4095x^{11}+3465x^{13} by 45045 to get \frac{1}{3}x^{3}+\frac{1}{5}x^{5}+\frac{1}{7}x^{7}+\frac{1}{9}x^{9}+\frac{1}{11}x^{11}+\frac{1}{13}x^{13}.

$$y=x+\frac{1}{3}x^{3}+\frac{1}{5}x^{5}+\frac{1}{7}x^{7}+\frac{1}{9}x^{9}+\frac{1}{11}x^{11}+\frac{1}{13}x^{13}$$