Solving a Linear Differential Equation: y'''+3yy''+2y'=3x+1

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SUMMARY

The equation y''' + 3yy'' + 2y' = 3x + 1 is not a linear differential equation due to the presence of the term 3yy'', which includes the product of the dependent variable y and its derivative y''. A linear ordinary differential equation (ODE) must adhere to the form where the dependent variable and its derivatives appear linearly. The general form of a linear ODE is given by {a_0}(x)(d^n y/dx^n) + {a_1}(x)(d^{n-1} y/dx^{n-1}) + ... + {a_n}(x)y = f(x), where the coefficients a_i(x) are functions of the independent variable only.

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blackblanx
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Is the following a linear differential equation? I think it is since the y near the y''.


y'''+3yy''+2y'=3x+1
 
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No y is a solution of the equation.

The general ordinary linear diff =ion is

{a_0}(x)\frac{{{d^n}y}}{{d{x^n}}} + {a_1}(x)\frac{{{d^{n - 1}}y}}{{d{x^{n - 1}}}} + {a_2}(x)\frac{{{d^{n - 2}}y}}{{d{x^{n - 2}}}}... + {a_n}(x)y = f(x)

You can have a y on the LHS only as shown
 
Last edited:
oh ok thank you
 

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