Find the specific solution of the differential equation dy/dx= 4y/x^2 with condition y(-4) = e.
dy/dx= 4y/x^2
dy/y = 4/x^2 dx
lny = -4/x +c
y = c e^(-4/x)
c e^(-4/-4) = c*e = e
So, c=1
y = e^(-4/x)
Sure, let's get solving! But first, might I ask, why did the differential equation go to therapy? It had some unresolved issues with its ex! Haha!
Ahem, now let's proceed. We have the differential equation dy/dx = 4y/x^2. To solve this, we can separate the variables and integrate:
dy/y = 4/x^2 dx
Integrating both sides gives us:
ln|y| = -4/x + C
Where C is the constant of integration. To proceed, we can exponentiate both sides to get rid of the natural logarithm:
|y| = e^(-4/x + C)
Since e is a positive number, we can drop the absolute value sign:
y = ±e^(-4/x + C)
Now, let's use the given initial condition y(-4) = e to find the value of the constant C. Substituting x = -4 and y = e into our equation, we have:
e = ±e^(-4/(-4) + C)
e = ±e^(1 + C)
Since e raised to any power is always positive, we can disregard the negative sign:
e = e^(1 + C)
To solve for C, we take the natural logarithm of both sides:
ln(e) = ln(e^(1 + C))
1 = (1 + C)
Therefore, C = 0.
Finally, substituting C = 0 back into our equation, we have:
y = e^(-4/x + 0)
y = e^(-4/x)
So, the specific solution of the given differential equation with the condition y(-4) = e is y = e^(-4/x). And there you have it! The solution is as specific as it gets, just like always knowing where to find your lost socks!
I hope this explanation brightened up your day a little!
To solve the differential equation dy/dx = 4y/x^2, we can separate the variables and integrate both sides.
Separating the variables:
dy/y = 4/x^2 * dx
Integrating both sides:
∫ (1/y) dy = ∫ (4/x^2) dx
ln|y| = ∫ (4/x^2) dx
To find the integral of (4/x^2), we can rewrite it as 4x^-2 and apply the power rule of integration:
ln|y| = -4x^-1 + C
where C is the constant of integration.
Now let's use the initial condition y(-4) = e.
Plugging in x = -4 and y = e into the equation, we get:
ln|e| = -4(-4)^-1 + C
ln(1) = -4(-1/4) + C
0 = 1 + C
Therefore, C = -1.
Substituting this value back into the equation, we get:
ln|y| = -4x^-1 -1
Using the property of logarithms, we can rewrite this equation as:
|y| = e^(-4x^-1 -1)
Now, we need to consider the absolute value. Since y(-4) = e is positive, we can remove the absolute value and get:
y = e^(-4x^-1 - 1)
This is the specific solution of the differential equation dy/dx = 4y/x^2 with the condition y(-4) = e.
To find the specific solution of the given differential equation dy/dx = 4y/x^2 with the initial condition y(-4) = e, we can use the method of separation of variables.
Here's how you can solve this differential equation:
Step 1: Rewrite the equation in separated form. Move all the y terms to the left side and all the x terms to the right side:
dy/y = 4/x^2 dx
Step 2: Integrate both sides with respect to their respective variables.
∫ (1/y) dy = ∫ (4/x^2) dx
Step 3: Evaluate the integrals.
ln|y| = -4/x + C
Step 4: Solve for y. Apply the exponential function to both sides to eliminate the natural logarithm:
y = e^(-4/x + C)
Step 5: Simplify the expression. Since e^C is just a constant, let's combine it with the constant e^(-4/x):
y = Ce^(-4/x)
Step 6: Apply the initial condition y(-4) = e. Substitute the given values into the equation:
e = Ce^(-4/(-4))
e = Ce^1
e = C
Step 7: Solve for the constant C.
Since e = C, the value of the constant is e.
Step 8: Substitute the constant into the simplified expression for y:
y = e^(-4/x)
Therefore, the specific solution of the given differential equation with the initial condition y(-4) = e is y = e^(-4/x).