# What is General solution: Definition and 311 Discussions

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form

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{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}+b(x)=0,}
where a0(x), …, an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, …, y(n) are the successive derivatives of an unknown function y of the variable x.
Such an equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives.
A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any.
The solutions of linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.

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1. ### Can I find a general solution to this circuit?

TL;DR Summary: I have to find an equivalent resistance of the circuit below, dependent on the amount of ##R_3## - resistors. Here is the circuit: I think there is no general solution. When I want to calculate it, I have to do...
2. ### Find the general solution of the given PDE

My take; ##ξ=-4x+6y## and ##η=6x+4y## it follows that, ##52u_ξ +10u=e^{x+2y}## for the homogenous part; we shall have the general solution; $$u_h=e^{\frac{-5}{26} ξ} f{η }$$ now we note that $$e^{x+2y}=e^{\frac{8ξ+η}{26}}$$ that is from solving the simultaneous equation; ##ξ=-4x+6y##...
3. ### I General solution of 1D vs 3D wave equations

For the 1 dimensional wave equation, $$\frac{\partial^2 u}{\partial x ^2} - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$ ##u## is of the form ##u(x \pm ct)## For the 3 dimensional wave equation however, $$\nabla ^2 u - \frac{1}{c^2}\frac{\partial ^2 u }{\partial t^2} = 0$$It appears...
4. ### MHB Help with a question (Bernoulli General solution)

Hello guys I hope you all are doing well. :) I found below question in a book by Martin Braun "Differential Equations and Their Applications An Introduction to Applied Mathematics (Fourth Edition)" The question : The Bernoulli differential equation is (dy/dt)+a(t)y=b(t)y^n. Multiplying through...
5. ### A Jackson Sec 2.6 on "general solution" of charge near sphere

Hi , I'd like a little bit of clarification about Section 2.6 from Jackson's classic book on E & M. Section 2.6 starts out with the problem of a "conducting sphere" near a point charge, but then it confusingly veers away to a problem where potential is prescribed to vary with azimuth and polar...
6. ### MHB Integrating Factor Method for Solving y' + y = e^{-2t}

$y'=-y+e^{(-2)t}$
7. ### I General solution of heat equation?

We know $$K(x,t) = \frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})$$ is a solution to the heat equation: $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ I would like to ask how to prove: $$u(x,t) = \int_{-\infty}^{\infty} K(x-y,t)f(y)dy$$ is also the solution to...

13. ### Finding the general solution of this nonhomogeneous linear system

Hi, I was trying to do the following problem. My attempt. Finding the reduced row echelon form for the system above. I do not see any way to proceed any further. The following is the solution presented in solution manual. How do I proceed to get the following answer?

47. ### I Finding Area of Graphs with x^2: A General Solution?

Hello, If I have an x^2 graph that goes from 0 to a point a. Is there a general solution to where the area of the left side is equal to the area of the right?
48. ### Using variation of parameters to derive a general solution?

Homework Statement "By choosing the lower limit of integration in Eq. (28) in the text as the initial point ##t_0##, show that ##Y(t)## becomes ##Y(t)=\int_{t_0}^t(\frac{y_1(s)y_2(t)-y_t(t)y_2(s)}{y_1(s)y_2'(s)-y_1'(s)y_2(s)})g(s)ds## Show that ##Y(t)## is a solution of the initial value...
49. ### Question on general solution to harmonic EoM

Homework Statement An equation of motion for a pendulum: (-g/L)sinΦ = Φ(double dot) Homework Equations L = length g = gravity ω = angular velocity Φο = initial Φ The Attempt at a Solution The solution is Φ=Asinωt+Bcosωt solving for A and B by setting Φ and Φ(dot) equal to zero respectively...
50. ### Obtaining General Solution of ODE

Homework Statement So they want me to obtain the general solution for this ODE. Homework Equations I have managed to turn it into d^2y/dx^2=(y/x)^2. The Attempt at a Solution My question is, can I simply make d^2y/dx^2 into (dy/dx)^2, cancel the power of 2 from both sides of the equation...