PYQs Based Test - 14 : Euler-Cauchy Equations

Consider the ordinary differential equation $x^2\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+2y=0.$ Given the values of y(1) = 0 and y(2) = 2, the value of y(3) (round off to 1 decimal place), is ________.

Consider the homogeneous ordinary differential equation $x^2\frac{d^{2}y}{d{x}^2}-3x\frac{dy}{dx}+3y=0,$ x > 0 with y(x) as a general solution. Given that y(1) = 1 and y(2) = 14 the value of y(1.5), rounded off to two decimal places, is ______.

The complete solution of the linear differential equation

$\frac{d^{2}z}{d{t}^2}+\left(p+q\right)\frac{dz}{dt}+pqz=0$

Consider two solutions x(t) = x₁(t) and x(t) = x₂(t) of the differential equation $\frac{d^{2}x\left(t\right)}{d{t}^2}+x\left(t\right)=0,t>0$, such that$x_1(0)=1,{\frac{d{x}_1\left(t\right)}{dt}|}_{t=0}=0,x_2\left(0\right)=0,{\frac{d{x}_2\left(t\right)}{dt}|}_{t=0}=1$

The Wronskian W≤ft( t ) = ≤ft| {*20c x_1≤ft( t )&x_2≤ft( t )\\ {{dx_1≤ft( t )}dt}&{{dx_2≤ft( t )}dt} } | at $t=\frac{\pi }{2}$ is

The solution of the differential equation $x^2\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y=\log x$ is