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The solution y(x) of the differential equation (x^3y^2 xy)dx=dy is. a) y(x)--->0 as x-->infinity b) y(x)-->infinity as x-->infinity C) y(x) decrease in [2, infinity),if y(0)=1 D) y(x) decrease in [2, infinity),ify(0)=-1?
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The solution y(x) of the differential equation (x^3y^2 xy)dx=dy is. a)...
Solution:
Given differential equation: (x^3y^2 + xy)dx = dy
To solve this differential equation, we can use the method of separation of variables.
Separating the variables, we have:
dy/dx = x^3y^2 + xy
Now, let's solve this differential equation step by step.
1. Homogeneous Differential Equation:
The given equation is a homogeneous differential equation. To verify this, we substitute y = vx, where v is a function of x.
dy/dx = d(vx)/dx = v + xdv/dx
Substituting these values in the original equation, we get:
v + xdv/dx = (x^3(v^2) + x)dx
Dividing both sides by x, we have:
v/x + dv/dx = x^2v^2 + 1
2. Linear Differential Equation:
The equation obtained after dividing by x is a linear differential equation. To solve this, we can use an integrating factor.
Rearranging the equation, we get:
dv/dx - x^2v^2 = -v/x + 1
The integrating factor for this linear differential equation is given by:
μ(x) = e^(∫(-x^2)dx) = e^(-x^3/3)
Multiplying both sides of the equation by the integrating factor, we get:
e^(-x^3/3)(dv/dx) - e^(-x^3/3)(x^2v) = -e^(-x^3/3)(v/x) + e^(-x^3/3)
The left-hand side can be simplified using the product rule of differentiation:
d/dx(e^(-x^3/3)v) = -e^(-x^3/3)(v/x) + e^(-x^3/3)
Therefore, the equation becomes:
d/dx(e^(-x^3/3)v) = 0
Integrating both sides with respect to x, we get:
e^(-x^3/3)v = C
where C is an arbitrary constant.
3. Final Solution:
Multiplying both sides by e^(x^3/3), we obtain:
v = Ce^(x^3/3)
Substituting y = vx, we get:
y = xCe^(x^3/3)
So, the final solution of the given differential equation is y(x) = xCe^(x^3/3), where C is an arbitrary constant.
Now, let's analyze the options:
a) y(x) approaches 0 as x approaches infinity:
From the solution, we can see that as x approaches infinity, the term e^(x^3/3) grows exponentially. Therefore, y(x) does not approach 0 as x approaches infinity. Hence, option (a) is incorrect.
b) y(x) approaches infinity as x approaches infinity:
As x approaches infinity, the term e^(x^3/3) grows exponentially. Therefore, y(x) also grows exponentially. Hence, option (b) is correct.
c) y(x) decreases in [2, infinity), if y(0) = 1:
To determine the behavior of y(x) in [2, infinity), we need more information about the constant C. The given initial condition y(
Given differential equation: (x^3y^2 + xy)dx = dy
To solve this differential equation, we can use the method of separation of variables.
Separating the variables, we have:
dy/dx = x^3y^2 + xy
Now, let's solve this differential equation step by step.
1. Homogeneous Differential Equation:
The given equation is a homogeneous differential equation. To verify this, we substitute y = vx, where v is a function of x.
dy/dx = d(vx)/dx = v + xdv/dx
Substituting these values in the original equation, we get:
v + xdv/dx = (x^3(v^2) + x)dx
Dividing both sides by x, we have:
v/x + dv/dx = x^2v^2 + 1
2. Linear Differential Equation:
The equation obtained after dividing by x is a linear differential equation. To solve this, we can use an integrating factor.
Rearranging the equation, we get:
dv/dx - x^2v^2 = -v/x + 1
The integrating factor for this linear differential equation is given by:
μ(x) = e^(∫(-x^2)dx) = e^(-x^3/3)
Multiplying both sides of the equation by the integrating factor, we get:
e^(-x^3/3)(dv/dx) - e^(-x^3/3)(x^2v) = -e^(-x^3/3)(v/x) + e^(-x^3/3)
The left-hand side can be simplified using the product rule of differentiation:
d/dx(e^(-x^3/3)v) = -e^(-x^3/3)(v/x) + e^(-x^3/3)
Therefore, the equation becomes:
d/dx(e^(-x^3/3)v) = 0
Integrating both sides with respect to x, we get:
e^(-x^3/3)v = C
where C is an arbitrary constant.
3. Final Solution:
Multiplying both sides by e^(x^3/3), we obtain:
v = Ce^(x^3/3)
Substituting y = vx, we get:
y = xCe^(x^3/3)
So, the final solution of the given differential equation is y(x) = xCe^(x^3/3), where C is an arbitrary constant.
Now, let's analyze the options:
a) y(x) approaches 0 as x approaches infinity:
From the solution, we can see that as x approaches infinity, the term e^(x^3/3) grows exponentially. Therefore, y(x) does not approach 0 as x approaches infinity. Hence, option (a) is incorrect.
b) y(x) approaches infinity as x approaches infinity:
As x approaches infinity, the term e^(x^3/3) grows exponentially. Therefore, y(x) also grows exponentially. Hence, option (b) is correct.
c) y(x) decreases in [2, infinity), if y(0) = 1:
To determine the behavior of y(x) in [2, infinity), we need more information about the constant C. The given initial condition y(
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The solution y(x) of the differential equation (x^3y^2 xy)dx=dy is. a) y(x)--->0 as x-->infinity b) y(x)-->infinity as x-->infinity C) y(x) decrease in [2, infinity),if y(0)=1 D) y(x) decrease in [2, infinity),ify(0)=-1?
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