Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$\begin{array} { l }
y = \frac { 1 } { 5 x } \int _ { 1 } ^ { 5 x } \frac { e ^ { t } } { t } d t \
x ^ { 2 } y ^ { \prime } + x y = \frac { 1 } { 5 } e ^ { 5 x }
\end{array}$\

Question

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$\begin{array} { l } 
y = \frac { 1 } { 5 x } \int _ { 1 } ^ { 5 x } \frac { e ^ { t } } { t } d t \
x ^ { 2 } y ^ { \prime } + x y = \frac { 1 } { 5 } e ^ { 5 x }
\end{array}$\

Quizplus · Accepted Answer

To check whether the given function is a solution or not, we need to substitute y = f(x) into the differential equation and see if it satisfies the equation.

Substituting y = f(x) into the differential equation, we get:

x^2y' + xy = (1/5)e^(5x)

x^2[d/dx (1/5x ∫(1 to 5x) e^t/t dt)] + x(1/5x ∫(1 to 5x) e^t/t dt) = (1/5)e^(5x)

Simplifying, we get:

(1/5) ∫(1 to 5x) e^t/t dt + ∫(1 to 5x) e^t/t dt + xe^(5x) = (1/5) e^(5x)

Combining the integrals on the left side, we get:

(6/5) ∫(1 to 5x) e^t/t dt + xe^(5x) = (1/5) e^(5x)

Taking the derivative with respect to x on both sides, we get:

(6/5) e^(5x) + 5xe^(5x) = e^(5x)

Simplifying, we get:

(6/5) e^(5x) + 5xe^(5x) - e^(5x) = 0

Combining like terms, we get:

(1/5) e^(5x) + 5xe^(5x) = 0

This equation is true for all x, which means that the given function y = f(x) is indeed a solution of the differential equation. Therefore, the answer is A.

Determine If the Given Function Y = F(x) Is a Solution