What is a solution to the linear differential equation: #dy/dx= y/x+x#?
# y = x^2 +Cx #
We have:
which we could type as:
which has the following form:
Thus, we are able to create an Integrating Factor:
Since this can now be divided, we can "separate the variables" to obtain:
Which is easy to incorporate:
Getting to the GS:
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Solve first the homogeneous equation that is separable:
Integrate both sides:
and taking the exponential of both sides:
Use now the variable coefficient method to find a solution to the compete equation in the form;
Differentiate using the product rule:
and substitute in the original equation:
and we do not need the constant because we can choose just one solution:
Then the complete solution of the equation is:
In fact:
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To solve the linear differential equation ( \frac{{dy}}{{dx}} = \frac{{y}}{{x}} + x ), you can use the method of integrating factors.

Rearrange the equation into standard linear form: [ \frac{{dy}}{{dx}}  \frac{{y}}{{x}} = x ]

Identify the integrating factor, denoted by ( \mu(x) ), which is calculated as: [ \mu(x) = e^{\int{\frac{1}{x}}dx} = e^{\lnx} = \frac{1}{x} ]

Multiply both sides of the equation by the integrating factor: [ \frac{{1}}{{x}}\frac{{dy}}{{dx}}  \frac{{1}}{{x}}\frac{{y}}{{x}} = 1 ]

Recognize that the lefthand side is the derivative of the product of the integrating factor and the dependent variable: [ \frac{{d}}{{dx}}\left( \frac{{y}}{{x}} \right) = 1 ]

Integrate both sides with respect to ( x ): [ \int{1 , dx} = \int{ \frac{{d}}{{dx}}\left( \frac{{y}}{{x}} \right) , dx} ] [ x + C_1 = \frac{{y}}{{x}} + C_2 ]

Solve for ( y ): [ y = xx + Cx ]
Where ( C ) is the constant of integration.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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