is the answer to find an equation of the curve that satisfies dy/dx=4x^3y and whose y-intercept is 7 y=7e^(x^4)
To find an equation of the curve that satisfies the given differential equation and has a y-intercept of 7, we can follow these steps:
Step 1: Start with the given differential equation:
dy/dx = 4x^3y
Step 2: Separate the variables by moving all the terms involving y to one side and all the terms involving x to the other side:
dy/y = 4x^3 dx
Step 3: Integrate both sides of the equation with respect to their respective variables:
∫ 1/y dy = ∫ 4x^3 dx
Step 4: Evaluate the integrals:
ln|y| = x^4 + C₁, where C₁ is the constant of integration.
Step 5: Solve for y by exponentiating both sides of the equation:
|y| = e^(x^4 + C₁)
Step 6: Rewrite e^(x^4 + C₁) as a product of two separate exponential functions using the properties of exponents:
|y| = e^(x^4) * e^(C₁)
Step 7: Introduce a new constant, C₂, where e^(C₁) = C₂:
|y| = C₂ * e^(x^4)
Step 8: Consider the absolute value to account for both positive and negative values of y:
y = ± C₂ * e^(x^4)
Step 9: Since the curve passes through the y-intercept of 7, substitute the coordinates (0, 7) into the equation to find the value of C₂:
7 = ± C₂ * e^(0^4)
7 = ± C₂
Since y-intercept is given as 7, C₂ must be positive. Therefore, we have:
y = C₂ * e^(x^4)
Given that 7 = C₂, the equation of the curve that satisfies the differential equation dy/dx = 4x^3y and the y-intercept is 7 is:
y = 7 * e^(x^4)