Integrating Factors Calculator

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Example

Created on 2024-06-20Asked by Camila Thomas (Solvelet student)

Find the integrating factor for the differential equation

\frac{dy}{dx} + 2y = 4x

Solution

To find the integrating factor for the differential equation

\frac{dy}{dx} + 2y = 4x

: The standard form of a linear first-order ordinary differential equation is:

\frac{dy}{dx} + P(x)y = Q(x)

Here,

P(x) = 2

and

Q(x) = 4x

. The integrating factor

\mu(x)

is given by:

\mu(x) = e^{\int P(x) \, dx}

Substitute

P(x) = 2

\mu(x) = e^{\int 2 \, dx}

\mu(x) = e^{2x}

Therefore, the integrating factor for the differential equation

\frac{dy}{dx} + 2y = 4x

is:

\mu(x) = e^{2x}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Sophia Harris on Solvelet

1. Find the integrating factor for the first-order linear differential equation

y' + 2xy = x

,2. Find the area under the curve

y = x^3 - x^2 + x

over the interval

[-1, 2]

using integration.,

DefinitionIntegrating factors are special functions used to make solving differential equations of the following type, dydx+P(x)y=Q(x) easier. Multiplying through by an integrating factor, typically a function e∫P(x)dx that is chosen so that it has the property of turning the left side of the equation into a derivative of a product of functions (a complementary property also held by the right side of the equation), allows the differential equation to be easily integrated. This process changes given equation to simple form and hence solve this equation. For instance, for the differential equation dy/dx +y = ex, the integrating factor is ex so that e∫ 1dx=ex. If we multiply through by ex, we get exdy/dx+exy=e2x so that d/dx(exy)=e2x. Taking the integral of both sides, we have exy=∫e2xdx.

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