General solution of differential equation ylogydx−xdy=0 is :
Get the complete, step-by-step math solution for: "General solution of differential equation y log y dx - x dy = 0 is :". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Separate the variables
The given differential equation is ylogydx−xdy=0. To solve this, we first rearrange the terms to separate the variables x and y. We move the x dy term to the right side and then divide both sides by x and ylogy to group dx with x and dy with y.
Step 2: Integrate both sides
Now that the variables are separated, we integrate both sides of the equation. The left side is a standard integral of 1/x, and the right side requires a substitution to solve.
Step 3: Solve the integrals
For the left side, ∫x1dx=log∣x∣. For the right side, let u=logy. Then du=y1dy. So, ∫ylogy1dy=∫u1du=log∣u∣=log∣logy∣. We add the constant of integration C to one side.
Step 4: Simplify the general solution
We can combine the logarithmic terms on the left side using the property loga−logb=log(a/b). Then, we convert the logarithmic equation to an exponential form. Let eC=A, where A is an arbitrary positive constant. This gives us the general solution.