Find the specific solution of the differential equation dy/dx = 4y/x^2 with condition y(−4) = e.
y= -1 - 4/x <- my answer
y=-1*e^(1/x)
y=e^(-4/x)
None of these
Nope. That condition y(-4) = e should have tipped you off that logs/exponentials were involved.
dy/dx = 4y/x^2
dy/y = 4/x^2
lny = -4/x + c
y = c*e^(-4x)
note that the two c values are different. We could have said that
lny = -4/x + lnc
y = e^(-4/x + lnc)
= e^(-4/x)*e^(lnc)
= c*e^(-4/x)
Now, y(-4) = e, so
ce^(-4/-4) = e
ce^1 = e
c = 1
y = e^(-4/x)
Did you check your answer?
y = -1 - 4/x
y' = 4/x^2
But you wanted y' = 4y/x^2
checking my answer,
y' = (4/x^2) e^(-4/x) = 4y/x^2
To find the specific solution of the given differential equation, we can separate the variables and integrate.
dy/dx = 4y/x^2
Rearranging the equation, we have:
(1/y)dy = (4/x^2)dx
Integrating both sides:
∫(1/y)dy = ∫(4/x^2)dx
ln|y| = -4/x + C
Exponentiating both sides:
|y| = e^(-4/x) * e^C
Since y(−4) = e, we know that y must be positive at x = -4. Therefore, we can drop the absolute value sign.
y = e^(-4/x) * e^C
To find the constant C, we substitute the initial condition y(−4) = e:
e = e^(-4/-4) * e^C
e = e^1 * e^C
This implies:
1 = e^C
Therefore, C = 0.
Substituting C = 0 back into the equation:
y = e^(-4/x)
Hence, the specific solution of the given differential equation with the condition y(−4) = e is y = e^(-4/x).
To find the specific solution of the given differential equation, we will use the method of separation of variables.
Given: dy/dx = 4y/x^2, and y(-4) = e.
Step 1: Separate the variables by putting all y terms on one side of the equation and all x terms on the other side.
dy/y = 4/x^2 dx
Step 2: Integrate both sides of the equation with respect to their respective variables.
∫(1/y) dy = ∫(4/x^2) dx
Step 3: Evaluate the integrals.
ln(|y|) = -4/x + C1,
where C1 is the constant of integration.
Step 4: Solve for y.
Taking the exponential of both sides, we have:
|y| = e^(-4/x + C1)
|y| = C2 * e^(-4/x),
where C2 = e^(C1) is a positive constant.
Step 5: Consider the initial condition y(-4) = e.
Substituting x = -4 and y = e into the equation, we have:
|e| = C2 * e^(-4/(-4)),
e = C2 * e,
C2 = 1.
Step 6: Determine the sign of y based on the absolute value.
Since C2 = 1, we have:
y = e^(-4/x)
Therefore, the specific solution to the given differential equation with the condition y(-4) = e is:
y = e^(-4/x).