ODE types solvable by two quadratures

The second order ordinary differential equation

d2ydx2=f(x,y,dydx) (1)

may in certain special cases be solved by using two quadraturesMathworldPlanetmath, sometimes also by reduction to a first order differential equationMathworldPlanetmath (http://planetmath.org/ODE) and a quadrature.

If the right hand side of (1) contains at most one of the quantities x, y and dydx, the general solution solution is obtained by two quadratures.

  • The equation

    d2ydx2=f(x) (2)

    is considered here (http://planetmath.org/EquationYFx).

  • The equation

    d2ydx2=f(y) (3)

    has as constant solutions all real roots of the equation  f(y)=0.  The other solutions can be gotten from the normal system

    dydx=z,dzdx=f(y) (4)

    of (3).  Dividing the equations (4) we get now  dzdy=f(y)z.  By separation of variablesMathworldPlanetmath and integration we may write


    whence the first equation of (4) reads


    here the variables and integrating give the general integral of (3) in the form

    dy2f(y)𝑑y+C1=x+C2. (5)

    The integration constant (http://planetmath.org/SolutionsOfOrdinaryDifferentialEquation) C1 has an influence on the form of the integral curves, but C2 only translates them in the direction of the x-axis.

  • The equation

    d2ydx2=f(dydx) (6)

    is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent3) with the normal system

    dydx=z,dzdx=f(z). (7)

    If the equation  f(z)=0  has real roots  z1,z2,,  these satisfy the latter of the equations (7), and thus, according to the former of them, the differential equation (6) has the solutions  y:=z1x+C1,  y:=z2x+C2,.

    The other solutions of (6) are obtained by separating the variables and integrating:

    x=dzf(z)+C. (8)

    If this antiderivative is expressible in closed form and if then the equation (8) can be solved for z, we may write


    Accordingly we have in this case the general solution of the ODE (6):

    y=g(x-C)𝑑x+C. (9)

    In other cases, we express also y as a function of z.  By the chain ruleMathworldPlanetmath, the normal system (7) yields




    Thus the general solution of (6) reads now in a parametric form as

    x=dzf(z)+C,y=zdzf(z)+C. (10)

    The equations 10 show that a translationPlanetmathPlanetmath of any integral curve yields another integral curve.

Title ODE types solvable by two quadratures
Canonical name ODETypesSolvableByTwoQuadratures
Date of creation 2015-03-20 17:04:58
Last modified on 2015-03-20 17:04:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Topic
Classification msc 34A05
Synonym second order ODE types solvable by quadratures
Related topic ODETypesReductibleToTheVariablesSeparableCase