IMO Practice Test: Differential Equations | Class 12 Mathematics

Question 1 of 6 hard

The particular solution of $\frac{dy}{dx}=\frac{y}{x}+\frac{\phi(y/x)}{\phi'(y/x)}$ is:

$\phi(y/x)=cx$

$\phi(y/x)=cy$

$\phi(y/x)=c$

$x\phi(y/x)=c$

Explanation: Substitute $v=y/x$: the equation reduces to $x\frac{dv}{dx}=\frac{\phi(v)}{\phi'(v)}$. Separating: $\frac{\phi'(v)}{\phi(v)}dv=\frac{dx}{x}$. Integrating: $\ln\phi(v)=\ln x+\ln C$, i.e. $\phi(y/x)=cx$. Hmm — actually $\phi(v)=cx$... so $\phi(y/x)=cx$. But let me choose option A.

Question 2 of 6 hard

Singular solution of $(y')^2-xy'+y=0$ (a Clairaut-type) is:

$y=x^2/4$

$y=x$

$y=cx-c^2$

$y=0$

Explanation: This is Clairaut's equation $y=xy'-y'^2$. The singular solution (envelope) is $x=2c$, $y=c^2$, eliminating $c$: $y=(x/2)^2=x^2/4$.

Question 3 of 6 hard

The solution of $\frac{dy}{dx}=\frac{x(2y-x)}{x(2x+y)}$ satisfies:

$\ln|x|+y/x+y^2/x^2=C$

Homogeneous — substitute $y=vx$

Both A and B

Neither

Explanation: RHS is $(2y/x-1)/(2+y/x)$ — depends on $y/x$. Homogeneous. Substitute $y=vx$.

Question 4 of 6 medium

Population growth model $dP/dt=kP$ gives, with $P(0)=P_0$:

$P=P_0e^{kt}$

$P=P_0+kt$

$P=P_0\ln(kt)$

$P=P_0e^{-kt}$

Explanation: Separable: $dP/P=k\,dt$. $\ln P=kt+C$. $P=P_0e^{kt}$ (exponential growth).

Question 5 of 6 medium

Newton's law of cooling: $dT/dt=-k(T-T_0)$. Solution with $T(0)=T_1$:

$T=T_0+(T_1-T_0)e^{-kt}$

$T=T_1e^{-kt}$

$T=T_0e^{-kt}$

$T=T_0+T_1e^{-kt}$

Explanation: Separable: $d(T-T_0)/(T-T_0)=-k\,dt$. $\ln(T-T_0)=-kt+C$. At $t=0$: $T-T_0=T_1-T_0$. $T=T_0+(T_1-T_0)e^{-kt}$.

Question 6 of 6 hard

Solving $x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y=0$ (Euler-Cauchy type), the general solution is:

$y=Ax+B/x$

$y=A\ln x+B$

$y=Ax^2+B$

$y=Ae^x+Be^{-x}$

Explanation: Euler equation. Substitute $x=e^t$: reduces to constant coefficient DE. Solution $y=c_1x+c_2x^{-1}=Ax+B/x$.

IMO Practice Test — Differential Equations