How to solve the seperable differential equation and when using the following initial condition: y(1)=2 ?

Question

Christopher Agresta · Accepted Answer

#y = root(6)(6ln(x)+6x+58)# #dy/dx = (1+x)/(xy^5)# #=> y^5dy = (1+x)/xdx=(1/x+1)dx# #=> inty^5dy = int(1/x+1)dx# #=> y^6/6 = ln(x)+x + C# #=> y^6 = 6ln(x)+6x + C# By the initial condition of #y(1)=2#, we have #2^6 = 6ln(1)+6+C = 6+C# #=> C = 2^6 - 6 = 58# #=> y^6 = 6ln(x)+6x+58# #:. y = root(6)(6ln(x)+6x+58)#

Elijah Abram · Answer

undefined To solve a separable differential equation with the initial condition y(1) = 2, follow these steps:

1. Separate the variables by moving all terms involving y to one side and terms involving x to the other side.
2. Integrate both sides of the equation with respect to their respective variables.
3. Solve for the constant of integration.
4. Apply the initial condition to find the specific value of the constant.
5. Substitute the value of the constant back into the solution to obtain the final solution.

If you provide the specific separable differential equation, I can demonstrate these steps for you.