Answer:
a solution is 1/2 *tan⁻¹ (2*y) = - tan⁻¹ (x²) + π/4
Step-by-step explanation:
for the equation
(1 + x⁴) dy + x*(1 + 4y²) dx = 0
(1 + x⁴) dy = - x*(1 + 4y²) dx
[1/(1 + 4y²)] dy = [-x/(1 + x⁴)] dx
∫[1/(1 + 4y²)] dy = ∫[-x/(1 + x⁴)] dx
now to solve each integral
I₁= ∫[1/(1 + 4y²)] dy = 1/2 *tan⁻¹ (2*y) + C₁
I₂= ∫[-x/(1 + x⁴)] dx
for u= x² → du=x*dx
I₂= ∫[-x/(1 + x⁴)] dx = -∫[1/(1 + u² )] du = - tan⁻¹ (u) +C₂ = - tan⁻¹ (x²) +C₂
then
1/2 *tan⁻¹ (2*y) = - tan⁻¹ (x²) +C
for y(x=1) = 0
1/2 *tan⁻¹ (2*0) = - tan⁻¹ (1²) +C
since tan⁻¹ (1²) for π/4+ π*N and tan⁻¹ (0) for π*N , we will choose for simplicity N=0 . hen an explicit solution would be
1/2 * 0 = - π/4 + C
C= π/4
therefore
1/2 *tan⁻¹ (2*y) = - tan⁻¹ (x²) + π/4
Answer:
yes that is correct. Do you have a question about it?
Answer:
8 grams
Step-by-step explanation:
The balance is in equilibrium, so the weights of the two sides are equal.
Let the weight of a square be s.
Left side: 2s + 4
Right side: s + 3(4) = s + 12
The weights are equal, so we set the two expressions equal.
2s + 4 = s + 12
s = 8
Answer: The weight of a square is 8 grams.
Answer:
B. 1 and -2
Explanation:
First, add up the equations:
x-x+3y+y=4
4y=4
y=1
Now, plug y into any of the original equations above and solve for x:
-x+1=3
-x=2
x=-2
Therefore, the answer is B. 1 and -2
