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
\[ y=C_1\cos 2x+C_2\sin 2x-\frac15\sin 3x \]2) Solve \(y'+y\tan x=\sin 2x\)
This is a first-order linear differential equation:
\[ \frac{dy}{dx}+Py=Q \] where \[ P=\tan x,\quad Q=\sin 2x \]Step 1: Integrating Factor
\[ IF=e^{\int Pdx} \] \[ IF=e^{\int \tan x\,dx} \] \[ IF=e^{\log\sec x} \] \[ IF=\sec x \]Step 2: Multiply throughout by IF
\[ \sec x\frac{dy}{dx}+y\sec x\tan x=\sin 2x\sec x \] The left-hand side becomes: \[ \frac{d}{dx}(y\sec x) \] Thus, \[ \frac{d}{dx}(y\sec x)=\sin 2x\sec x \] Since \[ \sin 2x=2\sin x\cos x \] we get \[ \sin 2x\sec x=2\sin x \] Hence, \[ \frac{d}{dx}(y\sec x)=2\sin x \]Step 3: Integrate
\[ y\sec x=\int 2\sin x\,dx \] \[ y\sec x=-2\cos x+C \] Multiplying by \(\cos x\), \[ y=-2\cos^2x+C\cos x \]3) Solve \(yy'+25x=0\)
Given:
\[ y\frac{dy}{dx}+25x=0 \] Rearranging, \[ y\frac{dy}{dx}=-25x \] Separate variables: \[ y\,dy=-25x\,dx \]Step 1: Integrate both sides
\[ \int y\,dy=\int -25x\,dx \] \[ \frac{y^2}{2}=-\frac{25x^2}{2}+C \] Multiplying by 2: \[ y^2=-25x^2+C \] or, \[ y^2+25x^2=C \]Conclusion
These examples demonstrate three important methods for solving differential equations: complementary function & particular integral, integrating factor, and separation of variables. Mastering these techniques helps in solving a wide variety of ordinary differential equations.
