Can someone please help?

Question

Can someone please help?

Solve the differential equation

dy/dx = 6xy
with the condition y(0) = 40

Find the solution to the equation y= ______

Answer 1

This equation is separable, so

dy/dx = 6xy
dy/y = 6xdx
Integrate both sides,
log(y)=3x²+C1
y=Ce^3x²
y(0)=40 -> C=40
therefore
y=40e^3x²

Answer 2

To solve the differential equation dy/dx = 6xy, we can use the method of separation of variables.

Step 1: Separate the variables

Start by rewriting the equation as dy/y = 6x*dx.

Step 2: Integrate both sides

Integrate both sides of the equation. We integrate the left side with respect to y, and the right side with respect to x.

∫ (1/y) dy = ∫ 6x dx

Step 3: Evaluate the integrals

The integral of (1/y) dy is ln|y| + C1, where C1 is a constant of integration.

The integral of 6x dx is 3x^2 + C2, where C2 is another constant of integration.

Step 4: Set up the general solution

Now we have ln|y| + C1 = 3x^2 + C2.

Step 5: Combine the constants

Combine the constants by replacing them with a single constant, C.

ln|y| = 3x^2 + C

Step 6: Solve for y

To solve for y, we take the exponential of both sides:

e^(ln|y|) = e^(3x^2 + C)

y = e^(3x^2 + C)

Step 7: Apply the initial condition

Finally, apply the initial condition y(0) = 40 to find the specific solution.

y(0) = e^(3(0)^2 + C) = e^C

We know that y(0) = 40, so e^C = 40.

Therefore, the specific solution to the differential equation dy/dx = 6xy with the condition y(0) = 40 is:

y = 40e^(3x^2)

So, the solution to the equation y = 40e^(3x^2)

Answer 3

To solve the given differential equation, we can use the method of separation of variables. Here's how we can do it step by step:

Step 1: Separate the variables
We start by separating the variables x and y on opposite sides of the equation. In this case, we move all terms involving y to the left side and all terms involving x to the right side:

1/y dy = 6x dx

Step 2: Integrate both sides
Next, we integrate both sides of the equation with respect to their respective variables. On the left side, integrate 1/y with respect to y. On the right side, integrate 6x with respect to x:

∫ (1/y) dy = ∫ 6x dx

Step 3: Evaluate the integrals
Integrating 1/y with respect to y gives us the natural logarithm (ln) of y:

ln|y| = 3x^2 + C

Integrating 6x with respect to x gives us 3x^2. We also introduce a constant of integration (C) on the right side of the equation, which represents an arbitrary constant.

Step 4: Solve for y
To solve for y, we can exponentiate both sides of the equation by taking the antilogarithm:

|y| = e^(3x^2 + C)

Since the absolute value of y can be either positive or negative, we can remove the absolute value by considering two cases:

1. For y > 0:
y = e^(3x^2 + C)

2. For y < 0:
y = -e^(3x^2 + C)

Step 5: Apply the initial condition
To find the particular solution that satisfies the initial condition y(0) = 40, substitute x = 0 and y = 40 into the equation from step 4. Let's consider the case where y > 0:

40 = e^(3(0)^2 + C)
40 = e^(0 + C)
40 = e^C

To evaluate the constant C, take the natural logarithm of both sides:

ln(40) = C

Substituting this value of C back into the equation from step 4, we get the final solution:

y = e^(3x^2 + ln(40))

So the solution to the given differential equation with the initial condition y(0) = 40 is:
y = e^(3x^2 + ln(40))

Note: The same process can be applied for the case where y < 0 by taking the negative value of the constant C when applying the initial condition.