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Solution
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Solution provided by AtoZmath.com
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Solve numerical differential equation using Euler method (2nd order derivative) calculator
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 1. Find y(1) for `y''=-4z-4y`, `x_0=0, y_0=0, z_0=1`, with step length 0.1
2. Find y(0.1) for `y''=1+2xy-x^2z`, `x_0=0, y_0=1, z_0=0`, with step length 0.1
3. Find y(0.2) for `y''=xz^2-y^2`, `x_0=0, y_0=1, z_0=0`, with step length 0.2
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Example1. Find y(0.1) for `y''=1+2xy-x^2z`, `x_0=0, y_0=1, z_0=0`, with step length 0.1 using Euler method (2nd order derivative)
Solution:
Given `y^('')=1+2xy-x^2z, y(0)=1, y'(0)=0, h=0.1, y(0.1)=?`
put `(dy)/(dx)=z` and differentiate w.r.t. x, we obtain `(d^2y)/(dx^2)=(dz)/(dx)`
We have system of equations
`(dy)/(dx)=z=f(x,y,z)`
`(dz)/(dx)=1+2xy-x^2z=g(x,y,z)`
Euler method for second order differential equation
`y_1=y_0+hf(x_0,y_0,z_0)=1+(0.1)*f(0,1,0)=1+(0.1)*(0)=1+(0)=1`
`:.y(0.1)=1`
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Input functions
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| Sr No. |
Function |
Input value |
| 1. |
`x^3` |
x^3 |
| 2. |
`sqrt(x)` |
sqrt(x) |
| 3. |
`root(3)(x)`
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root(3,x)
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| 4. |
sin(x) |
sin(x) |
| 5. |
cos(x) |
cos(x) |
| 6. |
tan(x) |
tan(x) |
| 7. |
sec(x) |
sec(x) |
| 8. |
cosec(x) |
csc(x) |
| 9. |
cot(x) |
cot(x) |
| 10. |
`sin^(-1)(x)` |
asin(x) |
| 11. |
`cos^(-1)(x)` |
acos(x) |
| 12. |
`tan^(-1)(x)` |
atan(x) |
| 13. |
`sin^2(x)` |
sin^2(x) |
| 14. |
`log_y(x)` |
log(y,x) |
| 15. |
`log_10(x)` |
log(x) |
| 16. |
`log_e(x)` |
ln(x) |
| 17. |
`e^x` |
exp(x) or e^x |
| 18. |
`e^(2x)` |
exp(2x) or e^(2x) |
| 19. |
`oo` |
inf |
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