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I am slightly stuck and was hoping someone could clarify about the superposition of solutions to linear, non-homogeneous differential equations.

I was taught that if you have two solutions to a linear equation, then any linear combination of these two solutions will also be a solution.

But surely this cannot be the case. Say I have the differential equation

$L(y)= a(x)\frac{d^2y}{x^2}+b(x)\frac{dy}{dx}+c(x)y=f(x)$ and I have two solutions $y_1$ and $y_2$, then the sum is not a solution as

$L(y_1+y_2)=L(y_1)+L(y_2)=2f(x)\neq f(x)$

I have a feeling that I have misunderstood something quite basic here...

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    Your are confusing iinear vs. [affine transformations.](https://en.wikipedia.org/wiki/Affine_transformation) – 2017-02-09

2 Answers 2

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You can add a solution to the homogeneous system to anything, Since that solution will give zero, f(x) remains unchanged.

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The space of solutions of a non-homogeneous linear differential equations is an affine space with direction the vector space of solutions of the associated homogeneous equation.

Indeed, if $y_1$ and $y_2$ are two solutions of the inhomogeneous equation, $y_1-y_2$ is a solution of $$a(x)y''(x)+by'(x)+c(x) y(x)=f(x)-f(x)=0 $$ Conversely, if y_2(x) is a solution of the inhomogeneous equation and $z(x)$ a solution of the associated homogeneous equation, it is easy to check $y_1(x)=y_2(x)+z(x)$ is a solution of the inhomogeneous equation.

Thus to completely solve a linear inhomogeneous differential equation you have to: 1) completely solve the linear homogeneous associated equation; 2) find one solution of the inhomogeneous equation; 3) add any solution of 1) to the particular solution of 2).

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    Thank you for your reply! I am aware of the method of solving inhomogeneous equations by the complementary function and the partcular integral, however what I am stuck on is that it seems that automatically if an equation is inhomogeneous it is non-linear. Take for example the statement in Wikipedia: 'A linear differential equation is called homogeneous if the following condition is satisfied: If $\phi (x)$ is a solution, so is $c\phi (x)$...A linear differential equation that fails this condition is called inhomogeneous' However is this condition not a defining feature of linear equation? – 2017-02-06
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    Perhpas I am confusing a linear operator with a linear equation? Is it perhaps the case that these equations are called linear because they contain only linear operators? – 2017-02-06
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    No it's not a defining feature of a linear equation but of a homogeneous equation (there exists homogeneous non-linear equations). An equation is linear if its homogeneous part is linear, which corresponds to a linear operator. For instance $\;\varphi\mapsto $a(x)\varphi''(x)+b(x)\varphi'(x)+c(x)\varphi(x)$ defines a linear operator on the space of twice differentiable function. – 2017-02-06