Solve the differential equation, we're using u substitution:
dy/dx= 4x + (4x/ sqr root (16-x^2))
Thanks. Please respond!
To solve the given differential equation by using u-substitution, follow these steps:
Step 1: Identify the function inside the square root as u.
Let u = 16 - x^2.
Step 2: Calculate du/dx.
Differentiate both sides of u = 16 - x^2 with respect to x.
du/dx = -2x.
Step 3: Solve for dx in terms of du.
Rearrange the equation in Step 2: du = -2x dx.
Solve for dx: dx = du / -2x.
Step 4: Substitute u and dx into the differential equation.
Substitute u and dx from Steps 1 and 3 into the given differential equation dy/dx = 4x + (4x / sqrt(16 - x^2)):
dy / (du / (-2x)) = 4x + (4x / sqrt(u)).
Multiply both sides by (-2x): -2x (dy / du) = 4x + (4x / sqrt(u)).
Step 5: Simplify the equation.
Rearrange the equation in Step 4: -2x (dy / du) = 4x (1 + (1 / sqrt(u))).
Divide by -2x on both sides: dy / du = -2 (1 + (1 / sqrt(u))).
Step 6: Separate and integrate the variables.
Separate variables: (1 + (1 / sqrt(u))) dy = -2 du.
Integrate both sides: ∫ (1 + (1 / sqrt(u))) dy = -2 ∫ du.
Step 7: Calculate the integrals.
The integral on the left side can be calculated as follows:
∫ (1 + (1 / sqrt(u))) dy = ∫ (1 + u^(-1/2)) dy.
= y + 2u^(1/2) + C, where C is the constant of integration.
The integral on the right side is simply:
-2 ∫ du = -2u + K, where K is another constant of integration.
Step 8: Combine the integrals.
Equating the two integrals gives:
y + 2u^(1/2) + C = -2u + K.
Step 9: Rearrange the equation to solve for y.
Solve for y:
y = -2u - 2u^(1/2) + (C - K).
Step 10: Substitute the value of u back in terms of x.
Recall that u = 16 - x^2:
y = -2(16 - x^2) - 2(16 - x^2)^(1/2) + (C - K).
Thus, the solution to the given differential equation is:
y = -2(16 - x^2) - 2(16 - x^2)^(1/2) + (C - K).
Please note that the constants C and K will be determined by any initial conditions or additional information given in the problem.