How to solve the separable differential equation and to the initial condition: y(0)=1 ?

Question

Samantha Adkison · Accepted Answer

Get the #x#s on one side and #y#s on the other, integrate, and simplify to get #y=sqrt(5/2(sqrt(x^2+1))-3/2)#.

We will begin to solve this first-order separable differential equation by separating it (no surprise there). If we add #8ysqrt(x^2+1)dy/dx# to both sides we get: #10x=8ysqrt(x^2+1)dy/dx# Now divide by #8sqrt(x^2+1)# to get: #(10x)/(8sqrt(x^2+1))=ydy/dx# Multiply by #dx# to finally end up with: #(10x)/(8sqrt(x^2+1))dx=ydy# Yay! We've separated the equation: we have #x# on one side and #y# on the other. The only thing that's left is to integrate both sides: #int(10x)/(8sqrt(x^2+1))dx=intydy# Let's start with the more complicated one of these: #int(10x)/(8sqrt(x^2+1))dx# First, take out a #10/8# and simplify: #5/4intx/(sqrt(x^2+1))dx# Now we can apply the substitution #u=x^2+1->(du)/dx=2x->du=2xdx# In order to apply the substitution, we need to multiply the inside of the integral by #2#, so we end up with #2x#. If we do that, we need to multiply the outside of the integral by #1/2#: #5/8int(2x)/(sqrt(x^2+1))dx# We can go ahead and substitute now: #5/8int(du)/sqrt(u)# Noticing that this is equivalent to #5/8intu^(-1/2)du# and using the reverse power rule, we end up with a solution of: #2*5/8(u^(1/2))+C=5/4sqrt(u)+C# Since #u=x^2+1#, we back-substitute to get our final answer: #5/4sqrt(x^2+1)+C# As for the other integral, #intydy#, well, that's just #y^2/2+C#. Alright, we've solved our integrals so we now have: #5/4sqrt(x^2+1)+C=y^2/2+C# Doing a little algebra to solve for #y# yields: #y=sqrt(5/2(sqrt(x^2+1))+C)# Note: Remember that manipulating the integration constant #C# makes no difference. Whatever we do to it, it's still a constant. Now we apply the initial condition #y(0)=1# to solve for #C#: #y=sqrt(5/2(sqrt(x^2+1))+C)# #1=sqrt(5/2(sqrt((0)^2+1))+C)# #1=5/2+C# #C=-3/2# Therefore our solution is #y=sqrt(5/2(sqrt(x^2+1))-3/2)#.

Samantha Adkison · Answer

undefined To solve a separable differential equation with the initial condition $ y(0) = 1 $, follow these steps:

1. Separate the variables by moving 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 using the initial condition $ y(0) = 1 $.
4. Substitute the constant of integration back into the equation to obtain the solution.

Would you like an example to illustrate these steps?