Solve the initial value problem explicity:
y'= 5xe^(-y), y(1)=1
This equation is separable:
y'/(5e^(-y)) = x
Integrate and substitute y(1)=1 to solve for the integration constant C.
See solution to your other post for reference:
http://www.jiskha.com/display.cgi?id=1296694049
To solve the initial value problem y' = 5xe^(-y) with the initial condition y(1) = 1 explicitly, we can use the method of separation of variables.
Step 1: Separate the variables
Since y' = 5xe^(-y), we can rewrite the equation as:
e^y dy = 5x dx
Step 2: Integrate both sides
Integrating both sides of the equation gives us:
∫ e^y dy = ∫ 5x dx
Using the integral rules, the left side becomes:
e^y = 5/2 x^2 + C
where C is the constant of integration.
Step 3: Solve for y
To solve for y, we need to isolate it in the equation. Taking the natural logarithm (ln) of both sides, we have:
ln(e^y) = ln(5/2 x^2 + C)
Since ln(e^y) simplifies to y, we get:
y = ln(5/2 x^2 + C)
Step 4: Apply the initial condition
To determine the value of the constant C, we can substitute the initial condition y(1) = 1 into the equation:
1 = ln(5/2 + C)
Now, we can solve for C. Take the exponential of both sides to get rid of the natural logarithm:
e^1 = 5/2 + C
This simplifies to:
e = 5/2 + C
Subtracting 5/2 from both sides gives us:
C = e - 5/2
Now we can substitute this value of C back into the equation derived in Step 3:
y = ln(5/2 x^2 + e - 5/2)
So, the explicit solution to the initial value problem y' = 5xe^(-y) with y(1) = 1 is given by:
y = ln(5/2 x^2 + e - 5/2)