How can I solve the differential equation?
C = 15ð(2Dh+(3/4)D²) + ë (94.25 Dh+35.34 D^2)/(pi D^2h/4)
first dC/dD?
second d^2C/dD^2?
first dC/dh?
second d^2C/dh^2?
To solve the given differential equation, we need to find the derivatives of C with respect to D and h. Let's go step by step.
1. First, let's find dC/dD (the first derivative of C with respect to D):
To differentiate C with respect to D, we treat h as a constant:
dC/dD = 15ð(2dh + (3/4)D^2)' + ë(94.25dh + 35.34D^2)' / (πD^2h/4)'
The derivative of the first term, 15ð(2dh + (3/4)D^2), with respect to D is:
= 15ð * (0 + (3/4) * 2D)
= 15ð * (3/2)D
= (45/2)ðD
The derivative of the second term, ë(94.25dh + 35.34D^2)/ (πD^2h/4), with respect to D is:
= ë * (0 + 35.34 * 2D) / (πD^2h/4)
= (70.68ëD) / (πD^2h/4)
= (70.68ëD) * (4 / πD^2h)
= (282.72ë) / (πDh)
Therefore, dC/dD = (45/2)ðD + (282.72ë) / (πDh)
2. Next, let's find d²C/dD² (the second derivative of C with respect to D):
To differentiate dC/dD with respect to D, we treat h as a constant:
d²C/dD² = [(45/2)ðD + (282.72ë) / (πDh)]'
The derivative of the first term, (45/2)ðD, with respect to D is simply (45/2)ð.
The derivative of the second term, (282.72ë) / (πDh), with respect to D is:
= [(282.72ë) / (πDh)] * (-1/(D^2))
= (-282.72ë) / (πD^3h)
Therefore, d²C/dD² = (45/2)ð - (282.72ë) / (πD^3h)
3. Moving on, let's find dC/dh (the first derivative of C with respect to h):
To differentiate C with respect to h, we treat D as a constant:
dC/dh = 15ð(2Dh + (3/4)D^2)' + ë(94.25Dh + 35.34D^2)' / (πD^2h/4)'
The derivative of the first term, 15ð(2Dh + (3/4)D^2), with respect to h is simply 15ð2D = 30ðD.
The derivative of the second term, ë(94.25Dh + 35.34D^2)/ (πD^2h/4), with respect to h is:
= ë(94.25D + 35.34D^2)' / (πD^2h/4)
= ë(94.25D + 35.34D^2)' * (4 / πD^2h)
= ë(35.34D) * (4 / πD^2h)
= (141.36ë) / (πDh)
Therefore, dC/dh = 30ðD + (141.36ë) / (πDh)
4. Lastly, let's find d²C/dh² (the second derivative of C with respect to h):
To differentiate dC/dh with respect to h, we treat D as a constant:
d²C/dh² = [30ðD + (141.36ë) / (πDh)]'
The derivative of the first term, 30ðD, with respect to h is 0 since D is constant.
The derivative of the second term, (141.36ë) / (πDh), with respect to h is:
= [(141.36ë) / (πDh)] * (-1/(h^2))
= (-141.36ë) / (πDh^3)
Therefore, d²C/dh² = (-141.36ë) / (πDh^3)
Now you have the first and second derivatives of C with respect to D and h.