Here, we show you a step-by-step solved example of linear differential equation. This solution was automatically generated by our smart calculator:
We can identify that the differential equation has the form: $\frac{dy}{dx} + P(x)\cdot y(x) = Q(x)$, so we can classify it as a linear first order differential equation, where $P(x)=2$ and $Q(x)=x$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$
Compute the integral
The integral of a constant is equal to the constant times the integral's variable
To find $\mu(x)$, we first need to calculate $\int P(x)dx$
So the integrating factor $\mu(x)$ is
Now, multiply all the terms in the differential equation by the integrating factor $\mu(x)$ and check if we can simplify
We can recognize that the left side of the differential equation consists of the derivative of the product of $\mu(x)\cdot y(x)$
Integrate both sides of the differential equation with respect to $dx$
Simplify the left side of the differential equation
We can solve the integral $\int xe^{2x}dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
We can solve the integral $\int e^{2x}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by finding the derivative of the equation above
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of the integral
The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
Now replace the values of $u$, $du$ and $v$ in the last formula
We can solve the integral $\int e^{2x}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by finding the derivative of the equation above
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of the integral
Multiplying fractions $-\frac{1}{2} \times \frac{1}{2}$
The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Solve the integral $\int xe^{2x}dx$ and replace the result in the differential equation
Multiplying the fraction by $e^{2x}x$
Any expression multiplied by $1$ is equal to itself
Multiplying the fraction by $e^{2x}$
Multiplying the fraction by $e^{2x}x$
Any expression multiplied by $1$ is equal to itself
Multiply the equation by the reciprocal of $e^{2x}$
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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