Question

Find a function y=f(x) satisfying the given differentialequation and the prescribed initial condition.

dy/dx= 5x⁴+4, y(0)=8

Answer 1

To find the function y=f(x) that satisfies the differential equation dy/dx = 5x⁴+4 and the initial condition y(0)=8, we integrate to get y = x⁵ + 4x + C. Using y(0) = 8, we find C and determine the final function is y = x⁵ + 4x + 8.

Step-by-step explanation:

To solve the differential equation dy/dx = 5x⁴ + 4 with the initial condition y(0) = 8, we need to integrate the right-hand side with respect to x. The general solution for y will include an integration constant, which is determined using the initial condition.

Integrating the right-hand side:

∫(5x⁴ + 4)dx = 5/5 * x⁵ + 4x + C = x⁵ + 4x + C

Now, applying the initial condition y(0) = 8:

8 = 0⁵ + 4(0) + C, which implies C = 8.

Therefore, the function that satisfies the differential equation and the initial condition is:

y = f(x) = x⁵ + 4x + 8.