What is an ODE y''+p(x)y'+q(x)y=r(x) Yp(x)=k₁cos(ωx)+k₂sin(ωx) α=min{a,b/k} an equation that contains a derivative of an unknown function, y(x)

an equation that contains a derivative of an unknown function, y(x)

How to transform a non-separable ODE to separable y(x)=C₁x^m₁+C₂x^m₂ You must use substitution for the part you cant separate, usually u=y/x x²y''+pxy'+qy=0 *coefficients same order as y* α=min{a,b/k}

You must use substitution for the part you cant separate, usually u=y/x

how to convert a first order non-linear ode to linear y'+p(x)y=0 let u=y¹⁻ⁿ, Create a new ODE using the equation u'+(1-n)p(x)u=(1-n)r(x) m²+(p-1)m+q=0 S(t)=S₀e^(rt)+(k/r)(e^(rt)-1), where S₀ is the initial investment, k is the investment per year, r is the interest rate, and t is time in years

let u=y¹⁻ⁿ, Create a new ODE using the equation u'+(1-n)p(x)u=(1-n)r(x)

how to solve an exact ODE` check exactness, find U(x,y) such that du/dx=M(x,y) or du/dy=N(x,y) remember to add K. Take the derivative of this U(x,y) with respect to whichever variable not used in the step before, compare this to du/dx or du/dy in step 2 to find K' and integrate if f(x,y) is continuous and bounded over a range R, the IVP has at least one solution over the interval |x-x0|<α x²y''+pxy'+qy=0 *coefficients same order as y* y'+p(x)y=0

check exactness, find U(x,y) such that du/dx=M(x,y) or du/dy=N(x,y) remember to add K. Take the derivative of this U(x,y) with respect to whichever variable not used in the step before, compare this to du/dx or du/dy in step 2 to find K' and integrate