Using separation of variables technique, solve the following differential equation with initial condition dy/dx = (yx + 5x) / (x^2 + 1) and y(3) = 5. The solution is:
a.) y^2 = ln(x^2 + 1) + 25 - ln10
b.) ln(abs(y+5)) = ln(x^2 + 1)
c.) ln(abs(y+5)) = arctan3 + ln10 - arctan3
d.) ln(abs(y+5)) = (1/2)ln(x^2 + 1) + (1/2)ln10
e.) y = ln(x^2 + 1) + 50 - ln10
Based on the initial condition y(3)=5, I know that e cannot be correct. However, I don't know where to start or how to work this out. My main problem is figuring out how to separate the variables.
I think D is your answer, but it's a strange way to write it. It just means
y+5 = sqrt(1) * sqrt(x^2+1)
Separation of variables here is rather easy, since you have
dy/dx = x(y+5)/(x^2+1) -- just divide by (y+5) and multiply by dx.
visit wolframalpha . com and type in
solve dy/dx = (yx + 5x) / (x^2 + 1), y(3)=5
it will solve the DE and if you click the Show Steps button, it will show how to separate the variables and all.
x(y+5) dx = (x^2+1) dy
x dx/(x^2+1) = dy/(y+5)
(1/2)ln(x^2+1) =
ln(y+5)+ c
if x = 3, y = 5
(1/2) ln 10 = ln 10 + c
c = -(1/2) ln 10
(1/2) ln(x^2 + 1) = ln(y+5) -(1/2) ln 10
(1/2) ln 10(x^2+1) = ln(y+5)
ln 10(x^2+1) = ln(y+5)^2
10(x^2+1) = (y+5)^2
Well, let's start by separating the variables in the differential equation. We have:
dy/dx = (yx + 5x) / (x^2 + 1)
To separate the variables, we can multiply both sides of the equation by (x^2 + 1) to get rid of the denominator on the right-hand side:
(x^2 + 1) dy = (yx + 5x) dx
Now, we can integrate both sides with respect to their respective variables.
∫(x^2 + 1) dy = ∫(yx + 5x) dx
Integrating the left-hand side gives us:
∫(x^2 + 1) dy = y∫(x^2 + 1) dx
= y(x^3/3 + x) + C1
On the right-hand side, we can write the integral as a sum of two integrals:
∫(yx + 5x) dx = ∫yx dx + ∫5x dx
Integrating each term separately:
= (y/2)x^2 + (5/2)x^2 + C2
Combining the two integrals on the right-hand side gives us:
= (y/2 + 5/2)x^2 + C2
Now, equating the two sides:
y(x^3/3 + x) + C1 = (y/2 + 5/2)x^2 + C2
At this point, we can consider the initial condition y(3) = 5. Plugging in x = 3 and y = 5 into the equation:
5(3^3/3 + 3) + C1 = (5/2 + 5/2)(3^2) + C2
Simplifying this:
5(9 + 3) + C1 = (5/2 + 5/2)(9) + C2
60 + C1 = 45 + C2
C1 - C2 = -15
Since we only need the expression in terms of y and x, we don't need to worry about the constant terms.
Therefore, the final expression is:
(y/2 + 5/2)x^2 = y(x^3/3 + x)
Now, let's take a look at the possible choices for the solution:
a.) y^2 = ln(x^2 + 1) + 25 - ln10
b.) ln(abs(y+5)) = ln(x^2 + 1)
c.) ln(abs(y+5)) = arctan3 + ln10 - arctan3
d.) ln(abs(y+5)) = (1/2)ln(x^2 + 1) + (1/2)ln10
e.) y = ln(x^2 + 1) + 50 - ln10
Comparing the given expression (y/2 + 5/2)x^2 = y(x^3/3 + x) to these choices, we can see that option b is the closest match:
ln(abs(y+5)) = ln(x^2 + 1)
So, the correct choice is:
b.) ln(abs(y+5)) = ln(x^2 + 1)
To solve the given differential equation using separation of variables, we want to separate the variables y and x by moving all terms containing y to one side and all terms containing x to the other side.
The given differential equation is:
dy/dx = (yx + 5x) / (x^2 + 1)
To separate the variables, we can multiply both sides of the equation by (x^2 + 1) to eliminate the denominator:
(x^2 + 1) dy/dx = yx + 5x
Now, let's move all terms containing y to one side:
(x^2 + 1) dy/dx - yx = 5x
Next, let's factor out dy/dx on the left side:
dy/dx (x^2 + 1 - yx) = 5x
To further separate the variables, we can divide both sides of the equation by (x^2 + 1 - yx):
dy/dx = 5x / (x^2 + 1 - yx)
Now, we have successfully separated the variables. To continue solving, we can multiply both sides of the equation by dx:
dy = (5x / (x^2 + 1 - yx)) dx
Next, we can integrate both sides of the equation. The left side will be the integral of dy, and the right side will be the integral of (5x / (x^2 + 1 - yx)) dx.
Integrating the left side, we have:
∫dy = ∫dx
On the right side, we need to recognize that the expression (x^2 + 1 - yx) is a factor of the denominator and the numerator, so we can use a substitution to simplify the integral.
Let's make the substitution u = x^2 + 1 - yx. Then, du = (2x - y) dx.
Now, the equation becomes:
∫1 dy = ∫(5x / u) du
Integrating both sides, we get:
y = 5 ∫(x/u) du
To solve the integral on the right side, we can separate it as:
y = 5 ∫(x/u) du = 5 ∫(x/u) (1/2x - y/2x) d(2x-y) = 5 ∫(x/u)(1/2x) d(2x) - 5 ∫(x/u)(y/2x) d(2x)
Simplifying, we have:
y = (5/2) ln|2x| - (5/2) ∫(y/u) d(2x)
Now, we can substitute back u = x^2 + 1 - yx:
y = (5/2) ln|2x| - (5/2) ∫(y/(x^2 + 1 - yx)) d(2x)
To continue solving, we can use the initial condition y(3) = 5 to evaluate the integral.
Plugging in x = 3 and y = 5, the equation becomes:
5 = (5/2) ln|2(3)| - (5/2) ∫(5/(3^2 + 1 - 5(3))) d(2(3))
Simplifying, we get:
5 = (5/2) ln(6) - (5/2) ∫(5/6) d(6)
The integral evaluates to:
5 = (5/2) ln(6) - (5/2) (5/6) (6)
Simplifying further, we have:
5 = (5/2) ln(6) - (5/2) (5/6) (6) = (5/2) ln(6) - 5/2
Now, we can isolate ln(6) by moving the constant term to the other side:
(5/2) ln(6) = 5 + 5/2
Multiplying both sides by 2/5, we get:
ln(6) = (10 + 5) / 2 * (2/5) = 3
Therefore, the solution to the differential equation with the initial condition y(3) = 5 is:
ln(6) = 3.
None of the answer choices provided match this result, so none of the given options are correct.
To solve the given differential equation using separation of variables technique, you need to follow these steps:
Step 1: Rewrite the equation.
dy/dx = (yx + 5x) / (x^2 + 1)
Step 2: Separate the variables.
dy / (yx + 5x) = dx / (x^2 + 1)
Step 3: Integrate both sides.
∫ (1 / (yx + 5x)) dy = ∫ (1 / (x^2 + 1)) dx
Step 4: Evaluate the integrals.
To integrate ∫ (1 / (yx + 5x)) dy, you need to perform a substitution. Let u = yx + 5x.
Then, du/dx = y + x(dy/dx), which rearranges to dy/dx = (du/dx - y) / x.
Substituting this into the differential equation, we have:
(dx/dx - y) / x = (u - y) / x = (yx + 5x - y) / x = (yx - y + 5x) / x = (yx - y)/x + 5
Therefore, the equation becomes:
dy/dx = (yx - y)/x + 5
Now, substitute u = yx + 5x into the equation:
du / dx - u / x = 5
Rearranging and simplifying, we get:
du / dx = u / x + 5
This equation is separable.
Step 5: Separate the variables.
du / (u + 5x) = dx / x
Step 6: Integrate both sides.
∫ 1 / (u + 5x) du = ∫ 1 / x dx
Step 7: Evaluate the integrals.
∫ 1 / (u + 5x) du = ln(|u + 5x|) + C1
∫ 1 / x dx = ln(|x|) + C2
Step 8: Solve for u.
ln(|u + 5x|) + C1 = ln(|x|) + C2
Taking exponents and combining constants, we have:
|u + 5x| = k|x|
Since the absolute value of a function is equal to a constant, we can have two possibilities:
u + 5x = kx OR u + 5x = -kx
Simplifying for u in each case:
u = (k - 5)x OR u = (-k - 5)x
Step 9: Solve for y.
Recall that u = yx + 5x.
Case 1: u = (k - 5)x
In this case, substitute u = (k - 5)x into the equation u = yx + 5x and solve for y:
(k - 5)x = yx + 5x
(k - 10)x = yx
y = (k - 10)
Case 2: u = (-k - 5)x
Following the same process, we obtain:
y = (-k - 10)
Step 10: Apply the initial condition.
We are given that y(3) = 5. Substituting this into the solutions obtained in Step 9, we can find the value of k.
For Case 1:
5 = k - 10
For Case 2:
5 = (-k - 10)
Solving these equations yields:
k = 15 for Case 1
k = -15 for Case 2
Step 11: Substitute k back into the solutions.
For Case 1: y = (15 - 10) = 5 (consistent with the initial condition)
For Case 2: y = (-(-15) - 10) = 5 (consistent with the initial condition)
Therefore, the solution to the differential equation with the initial condition y(3) = 5 is:
y = 5
So, the correct option is not among the choices provided.