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Chapter 8: Q79SE (page 359)

In Problems 25鈥�28 use (12) to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution.

\({x^2}\frac{{dy}}{{dx}} + xy = 10sinx;y = \frac{5}{x} + \frac{{10}}{x}\int_1^x {\frac{{sint}}{t}} dt\)

Short Answer

Expert verified

The indicated function is a solution of the differential function.

Step by step solution

01

Simplify the given differential equation.

Let the given differential equation be \(y = \frac{5}{x} + \frac{{10}}{x}\int_1^x {\frac{{sint}}{t}} dt\).

Multiply each side of the equation by \(x\).

\(\begin{array}{c}yx = 5 + 10\int_4^x {\frac{{sint}}{t}} \;dt\\yx - 5 = 10\int_4^x {\frac{{sint}}{t}} \;dt\end{array}\)

02

Determine the solution of the indicated function.

Take differential on both sides of the equation.

Multiply\(x\)on both sides of the equation.

\({x^2}\frac{{dy}}{{\;dx}} + yx = 10sinx\)

Hence, the indicated function is a solution of the differential function.

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