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Differential Equation (Bernoulli (\(y'+p(x)y=q(x)y^n\), \(n>2 || n<0\),…
Differential Equation
Bernoulli
\(y'+p(x)y=q(x)y^n\)
\(n>2 || n<0\)
\(v(t)=(1-n)y^{-n}y'\)
Constant Coefficients
Real & Distinct Roots
\(y(t)=c_1e^{k_1t}+c_2te^{k_2t}\)
Complex Roots
\(y(t)=c_1e^{at}cos(bt)+c_2e^{at}sin(bt)\)
Repeated Roots
\(y(t)=c_1e^{kt}+c_2te^{kt}\)
Reduction of Order
\(w=v'\)
\(w'=v''\)
\(y_2=vy_1\)
Undetermined Coefficients
Higher Order
\(y(t)=y_h(t)+y_p(t)\)
Guess
\(ke^{ax}\)
\(Ae^{ax}\)
\(kx^n\)
\(Ax^n+Bx^{n-1}+...+Jx+K\)
\(kcos(wx) \) & \( ksin(wx)\)
\(Acos(wx)+Bcos(wx)\)
\(ke^{ax}cos(wx)\) & \(ke^{ax}sin(wx)\)
\(e^{ax}(Acos(wx)+Bsin(wx))\)
Variation of Parameters
Higher Order
Short-cut method for \(y_p(t)\)
\(v'_1(t)y_1(t)+v'_2(t)y_2(t)\)=0
\(v'_1(t)y'_1(t)+v'_2(t)y'_2(t)\)=g(t)
\(y_p(t)=vy_h(t)\)
Laplace Transforms and Inverse
Higher Order
System of Differential Equations
Step Functions
Substitutions
\(v(x)=\frac{y}{x} \)
\(v(x)=ay-bx \)
Characteristic Equation
Homogeneous
Particular
Linear
\( y'+P(t)y=Q(t)\)
Exact
\(M(x,y)+N(x,y)\frac{dy}{dx}=0 \)
\(\psi_x=M(x,y)\)
\(\psi_y=N(x,y)\)
\(\psi_{xy}=\psi_{yx}\)
\(\psi=0\)
Series
Higher Order
Separable
\(N(y)\frac{dy}{dx}=M(x) \)
Slope Field
Drawing/Matching
Directional Fields
Intervals of Validity (VIP)
Wronskian/Cramer's rule