Second Order Linear ODE Summary

Second Order Linear D.E. Summary

Homogenous D.E.
y''+ay'+by=0.

Solve the Characteristic Equation: \lambda^2+a\lambda+b=0.
Case 1) Two real roots \lambda_1,\lambda_2: \implies \boxed{y=c_1e^{\lambda_1x}+c_2e^{\lambda_2x}}

Case 2) Real double root \lambda: \implies \boxed{y=c_1e^{\lambda x}+c_2xe^{\lambda x}}

Case 3) Complex Conjugate root \lambda_1,\lambda_2=-\frac{a}{2}\pm iw, where w=\sqrt{b-\frac{a^2}{4}}: \implies \boxed{y=e^{-\frac{a}{2}x}(c_1\cos wx+c_2\sin wx)}

Non-homogenous D.E.
General solution of non-homogenous D.E.: \displaystyle y=y_h+y_p, where y_h is the general solution of the homogenous equation, and y_p is the particular solution (with no arbitrary constants).

Method of Undetermined Coefficients (Guess and try method)
y''+p(x)y'+q(x)y=r(x).

Only works if r(x) is polynomial, exponential, sine or cosine (or sum/product of these).

Polynomial: Try y=Polynomial (e.g. y=Ax^2+Bx+C or y=Bx+C.)

Exponential (e^{kx}): Try y=ue^{kx}, where u is a function of x.

Trigonometric (\sin kx or \cos kx): Convert to complex differential equation by replacing y with z, replace \sin kx/\cos kx by e^{ikx}.

Try z=ue^{ikx}, where u is a function of x. After solving for z, take real/imaginary part of z for cosine/sine respectively.

Method of variation of parameters
y''+p(x)y'+q(x)y=r(x).

[Step 1)] Solve the homogenous D.E. y''+p(x)y'+q(x)y=0.

Get solution of the form y_h=c_1y_1+c_2y_2.

[Step 2)]
Let \displaystyle u=-\int\frac{y_2r}{W}\,dx and \displaystyle v=\int\frac{y_1r}{W}\,dx where W is the Wronskian \displaystyle W=y_1y_2'-y_1'y_2.

Particular solution: y_p=uy_1+vy_2.

General solution: y=y_h+y_p.

Forced Oscillations
Let F_0 be the amplitude of the driving (external) force. If F_0=0, by Newton’s Second Law, m\ddot{x}=-kx, hence \displaystyle \boxed{\ddot{x}=-\omega^2 x},
where \omega=\sqrt{k/m}. The value \omega is called the natural frequency.

If F_0\neq 0, then \displaystyle \boxed{m\ddot{x}+kx=F_0\cos\alpha t},
where \alpha is the driving (external) frequency.

At resonance (when \alpha=\omega), \displaystyle \boxed{x=\frac{F_0t}{2m\omega}\sin(\omega t)}.

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