Solving the Differential Equation ( frac{dy}{dx} 63y ) Using the Integrating Factor Method

In this article, we will solve the differential equation ( frac{dy}{dx} 63y ) using the integrating factor method. This method is particularly useful for solving linear first-order ordinary differential equations (ODEs). We will explore both the separation of variables approach and the integrating factor method.

Separation of Variables Method

Let's start with the separation of variables method. The given differential equation is:

ydxdy

To solve this, we can express it as:

dxdy16#x2062;3#x2062;y

This is a separable ordinary differential equation. We can separate the variables and integrate:

#x222B;y0yydy6#x2062;3#x2062;y#x222B;xxdx

This simplifies to:

163ln|y||))y0yyx#x2062;C

Exponentiating both sides, we get:

ye3x#x2062;C

Letting ( A e^{3C} ), the solution can be written as:

yA#x2062;e3x-2

Therefore, the solution is:

yC#x2062;e3x-2

Integrating Factor Method

For the integrating factor method, we start with the original equation:

y#x27;x63y

Expressing the equation in the standard form for a first-order linear ODE:

y#x27;x-63y0

Here, ( P(x) -63 ) and ( Q(x) 0 ).

Next, we find the integrating factor ( mu(x) ), defined as:

#x03BC;(x)e#x222B;xx-63#x2062;dx

This integral evaluates to:

#x03BC;(x)e-63x

Multiplying both sides of the original ODE by ( mu(x) ), we get:

e-63xy#x27;x-63e-63xy0

This simplifies to:

d{mspace{-5mu}overline{mspace{2mu}dmspace{2mu}},}(e-63xy)0

Integrating both sides, we get:

e-63xyC

Solving for ( y ), we obtain:

yCe63x

Applying the initial condition or simplifying further, we find:

yCe3x-2

Conclusion

In this article, we have explored two methods to solve the differential equation ( frac{dy}{dx} 63y ): the separation of variables method and the integrating factor method. Both methods lead to the same solution, demonstrating the versatility of these techniques in solving linear first-order ordinary differential equations.

Understanding these methods is crucial for anyone studying differential equations, especially when dealing with real-world applications in physics, engineering, and more. The integrating factor method is particularly powerful as it can be applied to a wide range of linear first-order ODEs.

If you need further assistance with differential equations or more advanced topics in calculus, feel free to explore the resources available online or seek help from your instructors.