Question:
Obtain a numerical solution of the differential equation:
![](https://static.wixstatic.com/media/20c892_7e1e494202d3454f879ab9dd5776572f~mv2.png/v1/fill/w_77,h_6,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_7e1e494202d3454f879ab9dd5776572f~mv2.png)
in the range 1.0(0.2)2.0 using the Taylor series method of order 4, given the initial conditions that
x = 1 when y = 4.
Answer:
![](https://static.wixstatic.com/media/20c892_52c725a2268c49bbb51a46d1c52b60c3~mv2.png/v1/fill/w_79,h_29,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_52c725a2268c49bbb51a46d1c52b60c3~mv2.png)
Use initial conditions to calculate the all values at their initial point:
![](https://static.wixstatic.com/media/20c892_77695f978cdd4570b386a49f4ddc3435~mv2.png/v1/fill/w_76,h_26,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_77695f978cdd4570b386a49f4ddc3435~mv2.png)
Use the Taylor series to find an approximation of y at x0+h:
![](https://static.wixstatic.com/media/20c892_6d945c34f747455aa60aa935401d92a3~mv2.png/v1/fill/w_83,h_45,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_6d945c34f747455aa60aa935401d92a3~mv2.png)
![](https://static.wixstatic.com/media/20c892_5f130b746b4442f8923738ae540b33e1~mv2.png/v1/fill/w_85,h_34,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_5f130b746b4442f8923738ae540b33e1~mv2.png)
At x = 1.6 the actual solution to the equation y'=3(1+x) - y is y = 5.348811636. Therefore, the percentage error of this approximation is:
![](https://static.wixstatic.com/media/20c892_205eb1cc245f49548ff270044c3fe02d~mv2.png/v1/fill/w_77,h_7,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_205eb1cc245f49548ff270044c3fe02d~mv2.png)
Therefore, in this case, the Taylor series approximation of order 4 gives a very accurate answer. The percentage error will be dependent in the order of Taylor series approximation; the higher the order, the smaller the error.