select each function y = f(x) that is a solution of the given differential equation.\n$\frac{dy}{dx}=30x^{4}$…

select each function y = f(x) that is a solution of the given differential equation.\n$\frac{dy}{dx}=30x^{4}$\nselect all solutions of $\frac{dy}{dx}=30x^{4}$.\n□ a. $y = 4x^{4}+30$\n□ b. $y = 6x^{5}+10$\n□ c. $y = 4x^{5}+8x$\n□ d. $y = 30x^{5}+18$

select each function y = f(x) that is a solution of the given differential equation.\n$\frac{dy}{dx}=30x^{4}$\nselect all solutions of $\frac{dy}{dx}=30x^{4}$.\n□ a. $y = 4x^{4}+30$\n□ b. $y = 6x^{5}+10$\n□ c. $y = 4x^{5}+8x$\n□ d. $y = 30x^{5}+18$

Answer

Explanation:

Step1: Recall the power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $\frac{dy}{dx}=nax^{n - 1}$.

Step2: Differentiate option A

If $y = 4x^{4}+30$, then $\frac{dy}{dx}=4\times4x^{3}=16x^{3}\neq30x^{4}$.

Step3: Differentiate option B

If $y = 6x^{5}+10$, then $\frac{dy}{dx}=5\times6x^{4}=30x^{4}$.

Step4: Differentiate option C

If $y = 4x^{5}+8x$, then $\frac{dy}{dx}=5\times4x^{4}+8 = 20x^{4}+8\neq30x^{4}$.

Step5: Differentiate option D

If $y = 30x^{5}+18$, then $\frac{dy}{dx}=5\times30x^{4}=150x^{4}\neq30x^{4}$.

Answer:

B. $y = 6x^{5}+10$