find an equation of the curve that satisfies dy/dx=4x^3y and whose y-intercept is 7
To find an equation of the curve that satisfies the given condition, we can use a technique called separation of variables.
Step 1: Separate the variables. Rewrite the equation by moving the terms involving y to one side and the terms involving x to the other side:
dy / y = 4x^3 dx
Step 2: Integrate both sides. Integrate both sides of the equation with respect to their respective variables:
∫(1/y) dy = ∫(4x^3) dx
Step 3: Evaluate the integrals. Integrate each side of the equation:
ln|y| = x^4 + C1
Step 4: Solve for y. Rewrite the equation in exponential form by taking the exponent of both sides:
|y| = e^(x^4 + C1)
Step 5: Add the constant. Since we know that the curve passes through the y-intercept (0, 7), substitute the values into the equation and solve for the constant:
7 = e^(0^4 + C1)
7 = e^C1
C1 = ln(7)
Step 6: Final equation. Substitute the value of C1 into the equation to obtain the final equation:
|y| = e^(x^4 + ln(7))
Therefore, the equation of the curve that satisfies dy/dx = 4x^3y and passes through the y-intercept (0, 7) is y = ±e^(x^4 + ln(7)).
To find an equation of the curve that satisfies the given differential equation and passes through the y-intercept (0, 7), we can use the technique of separating variables.
Step 1: Write the given differential equation in the form dy/dx = f(x)g(y).
dy/dx = 4x^3y
Step 2: Rearrange the equation by dividing both sides by y and multiplying both sides by dx.
dy/y = 4x^3 dx
Step 3: Integrate both sides with respect to their respective variables.
∫(1/y) dy = ∫(4x^3) dx
Integrating the left side yields:
ln|y| = x^4 + C1, where C1 is the constant of integration.
Step 4: Solve for y by taking the antilogarithm of both sides.
|y| = e^(x^4 + C1)
Since y-intercept occurs when x = 0, substitute these values into the equation to find the constant C1.
7 = e^(0^4 + C1)
7 = e^(C1)
Step 5: Determine the value of the constant C1.
Using the property that e^0 = 1, we have:
7 = 1 * e^C1
C1 = ln(7)
Step 6: Write the final equation in implicit form.
|y| = e^(x^4 + ln(7))
To simplify the equation, note that |y| = ±y:
y = ±e^(x^4 + ln(7))
Therefore, the equation of the curve that satisfies dy/dx = 4x^3y and passes through the y-intercept (0, 7) is y = ±e^(x^4 + ln(7)).