Solving the Differential Equation ( y'' cxy ) Where ( c ) is a Constant
When dealing with differential equations, particularly those involving a product of the dependent variable and its argument, such as ( y'' cxy ), various methods can be applied, including the power series method and direct integration. This article explores the solution process for this differential equation and discusses the methods used to solve it.
Power Series Method
To solve the differential equation ( y'' cxy ) using the power series method, we start by assuming a power series solution of the form:
[ y A Bx Cx^2 Dx^3 cdots ]
Similarly, the first and second derivatives of ( y ) are:
[ y' B 2Cx 3Dx^2 4Ex^3 cdots ]
[ y'' 2C 6Dx 12Ex^2 20Fx^3 cdots ]
Next, we consider the term ( cxy ):
[ cxy Acx Bcx^2 Ccx^3 Dcx^4 cdots ]
Subtracting ( cxy ) from ( y'' ), we get:
[ y'' - cxy 2C (6D - Ac)x (12E - Bc)x^2 (20F - Cc)x^3 cdots ]
Given the initial conditions ( y(0) A ) and ( y'(0) B ), we can determine the coefficients step by step:
( C 0, D frac{A}{6}, E frac{B}{12}, F 0, G frac{Dc}{30} frac{Ac^2}{180} ), and so on.
The general solution can be derived from the series expansion and the coefficients found.
Direct Integration Method
Another approach to solving ( y'' cxy ) is to use direct integration. Starting with the differential equation:
[ frac{dy}{dx} cxy ]
Divide both sides by ( y ) and integrate:
[ int frac{1}{y} , dy int cx , dx ]
This leads to:
[ ln |y| frac{cx^2}{2} C_1 ]
Exponentiating both sides, we obtain:
[ y e^{frac{cx^2}{2} C_1} A e^{frac{cx^2}{2}} ]
Where ( A e^{C_1} ) is a constant. This is the general solution to the differential equation.
To verify this solution, we substitute it back into the original differential equation:
[ frac{d}{dx} left( A e^{frac{cx^2}{2}} right) c x A e^{frac{cx^2}{2}} ]
The left side simplifies to:
[ A frac{d}{dx} left( e^{frac{cx^2}{2}} right) A c x e^{frac{cx^2}{2}} ]
This matches the right side exactly, confirming that ( y A e^{frac{cx^2}{2}} ) is indeed the solution.
Conclusion
The differential equation ( y'' cxy ) can be solved using both the power series method and direct integration. The power series approach involves finding the coefficients of the series, while the direct integration method simplifies the process to a straightforward exponential solution. Both methods are valuable tools in solving such differential equations.