Solve for D J=p*(D^4-(D-2w)^4)/32

Problem

J=(p*(D4−(D−2*w)4))/32

Solution

1. Multiply both sides by 32 and divide by p to isolate the terms containing D

(32*J)/p=D4−(D−2*w)4

2. Expand the binomial (D−2*w)4 using the binomial theorem.

(D−2*w)4=D4−4*D3*(2*w)+6*D2*(2*w)2−4*D*(2*w)3+(2*w)4

(D−2*w)4=D4−8*w*D3+24*w2*D2−32*w3*D+16*w4

3. Substitute the expansion back into the equation and simplify by subtracting D4

(32*J)/p=D4−(D4−8*w*D3+24*w2*D2−32*w3*D+16*w4)

(32*J)/p=8*w*D3−24*w2*D2+32*w3*D−16*w4

4. Divide the entire equation by 8*w to simplify the coefficients.

(4*J)/(p*w)=D3−3*w*D2+4*w2*D−2*w3

5. Rearrange the equation into a standard polynomial form set to zero.

D3−3*w*D2+4*w2*D−(2*w3+(4*J)/(p*w))=0

6. Solve for D using the cubic formula or numerical methods, as this is a general cubic equation in terms of D

Final Answer

D3−3*w*D2+4*w2*D−(2*w3+(4*J)/(p*w))=0

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