Some Questions on Differential Equations
Differential Equations: Detailed Step-by-Step Solutions In this article, we solve three important differential equations step by step using standard methods such as complementary function and particular integral , integrating factor , and separation of variables . 1) Solve \(y''+4y=\sin(3x)\) This is a linear differential equation with constant coefficients: \[ y''+4y=\sin(3x) \] The general solution is: \[ y=y_c+y_p \] where \(y_c\) is the complementary function and \(y_p\) is the particular integral. Step 1: Complementary Function The auxiliary equation is: \[ m^2+4=0 \] \[ m^2=-4 \] \[ m=\pm 2i \] Therefore, the complementary function is: \[ y_c=C_1\cos 2x+C_2\sin 2x \] Step 2: Particular Integral \[ (D^2+4)y=\sin 3x \] \[ y_p=\frac{1}{D^2+4}\sin 3x \] Using: \[ f(D)\sin ax=f(-a^2)\sin ax \] \[ y_p=\frac{1}{-9+4}\sin 3x \] \[ y_p=-\frac15\sin 3x \] Step 3: General Solution \[ y=C_1\cos 2x+C_2\sin 2x-\frac15\sin 3x \] Fi...