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Solution
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Solution provided by AtoZmath.com
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Solve numerical differential equation using Runge-Kutta 2 method (second order differential equation) 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(0.2) for `y''=1+2xy-x^2z`, `x_0=0, y_0=1, z_0=0`, with step length 0.1 using Runge-Kutta 2 method (second order differential equation) Solution:Given `y^('')=1+2xy-x^2z, y(0)=1, y'(0)=0, h=0.1, y(0.2)=?` 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)` Method-1 : Using formula `k_2=hf(x_0+h,y_0+k_1,z_0+l_1)`Second order Runge-Kutta (RK2) method for second order differential equation formula `k_1=hf(x_n,y_n,z_n)` `l_1=hg(x_n,y_n,z_n)` `k_2=hf(x_n+h,y_n+k_1,z_n+l_1)` `l_2=hg(x_n+h,y_n+k_1,z_n+l_1)` `y_(n+1)=y_n+(k_1+k_2)/2` `z_(n+1)=z_n+(l_1+l_2)/2`
for `n=0,x_0=0,y_0=1,z_0=0` `k_1=hf(x_0,y_0,z_0)` `=(0.1)*f(0,1,0)` `=(0.1)*(0)` `=0` `l_1=hg(x_0,y_0,z_0)` `=(0.1)*g(0,1,0)` `=(0.1)*(1)` `=0.1` `k_2=hf(x_0+h,y_0+k_1,z_0+l_1)` `=(0.1)*f(0.1,1,0.1)` `=(0.1)*(0.1)` `=0.01` `l_2=hg(x_0+h,y_0+k_1,z_0+l_1)` `=(0.1)*g(0.1,1,0.1)` `=(0.1)*(1.199)` `=0.1199` `y_1=y_0+(k_1+k_2)/2` `=1+0.005` `=1.005` `z_1=z_0+(l_1+l_2)/2` `=0+0.11` `=0.11` `x_1=x_0+h=0+0.1=0.1`
for `n=1,x_1=0.1,y_1=1.005,z_1=0.11` `k_1=hf(x_1,y_1,z_1)` `=(0.1)*f(0.1,1.005,0.11)` `=(0.1)*(0.11)` `=0.011` `l_1=hg(x_1,y_1,z_1)` `=(0.1)*g(0.1,1.005,0.11)` `=(0.1)*(1.1999)` `=0.12` `k_2=hf(x_1+h,y_1+k_1,z_1+l_1)` `=(0.1)*f(0.2,1.016,0.2299)` `=(0.1)*(0.2299)` `=0.023` `l_2=hg(x_1+h,y_1+k_1,z_1+l_1)` `=(0.1)*g(0.2,1.016,0.2299)` `=(0.1)*(1.3972)` `=0.1397` `y_2=y_1+(k_1+k_2)/2` `=1.005+0.017` `=1.022` `x_2=x_1+h=0.1+0.1=0.2` `:.y(0.2)=1.022` | `n` | `x_n` | `y_n` | `z_n` | `k_1` | `l_1` | `k_2` | `l_2` | `x_(n+1)` | `y_(n+1)` | `z_(n+1)` | | 0 | 0 | 1 | 0 | 0 | 0.1 | 0.01 | 0.1199 | 0.1 | 1.005 | 0.11 | | 1 | 0.1 | 1.005 | 0.11 | 0.011 | 0.12 | 0.023 | 0.1397 | 0.2 | 1.022 | |
Method-2 : Using formula `k_2=hf(x_0+h/2,y_0+k_1/2,z_0+l_1/2)`Second order Runge-Kutta (RK2) method for second order differential equation formula `k_1=hf(x_n,y_n,z_n)` `l_1=hg(x_n,y_n,z_n)` `k_2=hf(x_n+h/2,y_n+k_1/2,z_n+l_1/2)` `l_2=hg(x_n+h/2,y_n+k_1/2,z_n+l_1/2)` `y_(n+1)=y_n+k_2` `z_(n+1)=z_n+l_2`
for `n=0,x_0=0,y_0=1,z_0=0` `k_1=hf(x_0,y_0,z_0)` `=(0.1)*f(0,1,0)` `=(0.1)*(0)` `=0` `l_1=hg(x_0,y_0,z_0)` `=(0.1)*g(0,1,0)` `=(0.1)*(1)` `=0.1` `k_2=hf(x_0+h/2,y_0+k_1/2,z_0+l_1/2)` `=(0.1)*f(0.05,1,0.05)` `=(0.1)*(0.05)` `=0.005` `l_2=hg(x_0+h/2,y_0+k_1/2,z_0+l_1/2)` `=(0.1)*g(0.05,1,0.05)` `=(0.1)*(1.0999)` `=0.11` `y_1=y_0+k_2` `=1+0.005` `=1.005` `z_1=z_0+l_2` `=0+0.11` `=0.11` `x_1=x_0+h=0+0.1=0.1`
for `n=1,x_1=0.1,y_1=1.005,z_1=0.11` `k_1=hf(x_1,y_1,z_1)` `=(0.1)*f(0.1,1.005,0.11)` `=(0.1)*(0.11)` `=0.011` `l_1=hg(x_1,y_1,z_1)` `=(0.1)*g(0.1,1.005,0.11)` `=(0.1)*(1.1999)` `=0.12` `k_2=hf(x_1+h/2,y_1+k_1/2,z_1+l_1/2)` `=(0.1)*f(0.15,1.0105,0.17)` `=(0.1)*(0.17)` `=0.017` `l_2=hg(x_1+h/2,y_1+k_1/2,z_1+l_1/2)` `=(0.1)*g(0.15,1.0105,0.17)` `=(0.1)*(1.2993)` `=0.1299` `y_2=y_1+k_2` `=1.005+0.017` `=1.022` `x_2=x_1+h=0.1+0.1=0.2` `:.y(0.2)=1.022` | `n` | `x_n` | `y_n` | `z_n` | `k_1` | `l_1` | `k_2` | `l_2` | `x_(n+1)` | `y_(n+1)` | `z_(n+1)` | | 0 | 0 | 1 | 0 | 0 | 0.1 | 0.005 | 0.11 | 0.1 | 1.005 | 0.11 | | 1 | 0.1 | 1.005 | 0.11 | 0.011 | 0.12 | 0.017 | 0.1299 | 0.2 | 1.022 | |
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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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